Matrix 1

Installation Guide

2001 Directed Electronics, Inc. N909429 8-01

Rev. N/C 1.1

The Bitwriter (p/n 998T) requires chip version 1.4 or newer to program this unit.
Bitwriter, Code Hopping, DEI, Directed, Doubleguard, ESP, FailSafe, Ghost Switch, Learn Routine, Nite-Lite, Nuisance Prevention Circuitry, NPC, Revenger, Silent Mode, Soft Chirp, Stinger, Valet, Vehicle Recovery System, VRS, and Warn Away are all Trademarks or Registered Trademarks of Directed Electronics, Inc., Vista, California.

Table of Contents

What Is Included..4 Primary Harness (H1) Wire Connection Guide..4 Primary Harness Wiring Diagram.4 Primary Harness Wiring Guide..4 Door Lock Harness (H2) Wire Connection Guide..9 Plug-In Harnesses..10 I Super Bright LED.10 Valet/Program Switch..10 Programmer Interface..10 On-Board Dual Stage Shock Sensor.11 B Optional Sensor Harness, 4-Pin Connector.11 P Programming Jumper..12 Light Flash Jumper..12 Bypassing Sensor Inputs..12 System Features Learn Routine. 13 System Features Menus.. 15 Menu #1-Basic Features..15 Menu #2-Advanced Features..16 Feature Descriptions.. 16 Menu #1 - Basic Features..16 Menu #2 - Advanced Features..18 Transmitter/Receiver Learn Routine.19 Transmitter Configurations..22 3-Button Transmitter Configuration.22 4-Button Transmitter Configuration.22 Optional Radar Master Transmitter.22 Multi-Level Security Arming..23 L Smart Power Up II..23 Table of Zones..24 Long Term Event History..24 Optional Vehicle Recovery System. 25 To Arm the VRS..25 To Disarm the VRS..25 False Alarm Control Technology (FACT).26 Troubleshooting... 27 Wiring Quick Reference Guide.28

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What Is Included
One control module with on-board shock sensor One XHF receiver/antenna with harness Two three-button remote transmitters One 514C siren One 12-pin primary harness with starter kill One plug-in Valet/Program switch
One plug-in LED indicator with bezel One 3-pin door lock harness Two window decals One patent card One warranty registration One installation guide One owners guide
Primary Harness (H1) Wire Connection Guide
Primary Harness Wiring Diagram
H1/1 H1/2 H1/3 H1/4 H1/5 H1/6 H1/7 H1/8 H1/9 H1/10 H1/11 H1/12
___ ORANGE ___ WHITE ___ WHITE/BLUE ___ BLACK/WHITE ___ GREEN ___ BLUE ___ VIOLET ___ BLACK ___ YELLOW ___ BROWN ___ RED ___ RED/WHITE
(-) 500 mA Ground-When-Armed Output (+)/(-) Selectable Light Flash Output (-) 200 mA Channel 3 Programmable Output (-) 200 mA Domelight Supervision Output (-) Door Trigger Input, Zone 3 (-) Instant Trigger Input, Zone 1 (+) Door Trigger Input, Zone 3 (-) Chassis Ground Input (+) Switched Ignition Input, Zone 5 (+) Siren Output (+) Constant Power Input (-) 200 mA Channel 2 Output
Primary Harness Wiring Guide
This guide describes in detail the connection of each wire. Also included are possible applications of each wire. This system was designed with the ultimate in flexibility and security in mind. Many of the wires have more than one possible function. Please read the instructions carefully to ensure a thorough understanding of this unit and how it operates. 4
2001 Directed Electronics, Inc.
H1/1 ORANGE (-) ground-when-armed output This wire supplies a (-) ground as long as the system is armed. This output ceases as soon as the system is disarmed. The orange wire is pre-wired to control the P/N 8618 starter kill relay. It can supply up to 500 mA of current. NOTE: If connecting the H1/1 ORANGE wire to control another module, such as a P/N 529T or P/N 530T window module, a 1 amp diode (type 1N4004) will be required. (See the following diagram.)
IMPORTANT! Never interrupt any wire other than the starter wire.
H1/2 WHITE (+/-) selectable light flash output As shipped, the H1/2 WHITE wire should be connected to the (+) parking light wire. If the light flash polarity jumper near the main plug is moved to the opposite position (see the Programming Jumper section of this guide), this wire supplies a (-) 200 mA output. This is suitable for driving (-) light control wires in Toyota, Lexus, BMW, some Mitsubishi, some Mazda, and other models.

H1/3 WHITE/BLUE 200 mA (-) channel 3 programmable output This wire provides a (-) 200 mA output whenever the transmitter button(s) controlling channel three is pressed. This output can be programmed to provide the following types of output (see System Features Learn Routine section of this guide):
A validity output will send a signal as long as the transmission is received. A latched output will send a continuous signal after the button controlling channel three is pressed and released. The signal will continue until the button controlling channel three is pressed again. A latched/reset with ignition output works similar to the latched output, but will also reset (output will stop) when the ignition is turned on and then off. This output can also be shut off at any time by pressing the transmitter button that controls Channel 3 again. s A 30-second timed output will send a signal for 30 seconds when channel three is pressed. This output can be shut off during the 30-second period by pressing the Channel 3 button again. This output can also be programmed to provide a second unlock pulse when the disarm button is pressed within 15 seconds after disarming the system. This can be used to unlock the passenger doors when installing progressive door locks.
IMPORTANT! Never use this wire to drive anything except a relay or a low-current input! This transistorized output can only supply 200 mA, and connecting directly to a solenoid, motor, or other high-current device will cause the module to fail.
H1/4 BLACK/WHITE (-) 200 mA domelight supervision output Connect this wire to the optional domelight supervision relay as shown below:
IMPORTANT! This output is only intended to drive a relay. It cannot be connected directly to the domelight circuit, as the output cannot support the current draw of one or more bulbs.
H1/5 GREEN (-) door trigger input, zone 3 Most vehicles use negative door trigger circuits. Connect the green wire to a wire showing ground when any door is opened. In vehicles with factory delays on the domelight circuit, there is usually a wire unaffected by the delay circuitry. This wire will report Zone 3.
H1/6 BLUE (-) instant trigger, zone 1 This input will respond to a negative input with an instant trigger. It is ideal for hood and trunk pins and will report on Zone 1. It can also be used with the P/N 506T Glass Breakage Sensor, as well as other Directed single stage sensors. The H1/6 BLUE instant trigger wire can be used to shunt sensors during operation, using the auxiliary channels. When any of the auxiliary channels are transmitted, the H1/6 BLUE wire monitors for a ground. If a ground is detected within 5 seconds of transmission, the sensors and the instant trigger input on the BLUE wire will be shunted until 5 seconds after the ground is removed. This allows the customer to access the trunk, remote start the vehicle or roll the windows down without first disarming the alarm. (See Bypassing Sensor Inputs section of this guide.) H1/7 VIOLET (+) door trigger input, zone 3 This type of dome circuit is used in many Ford products. Connect the violet wire to a wire that shows (+)12V when any door is opened, and ground when the door is closed. This wire will report Zone 3.

H1/8 BLACK (-) chassis ground connection Remove any paint and connect this wire to bare metal, preferably with a factory bolt rather than your own screw. (Screws tend to either strip or loosen with time.) We recommend grounding all your components, including the siren, to the same point in the vehicle.
H1/9 YELLOW (+) ignition input, zone 5 Connect this wire to the (+)12V ignition wire. This wire is pre-wired to the starter kill relay and must show (+)12V with the key in Run position and during cranking. Take great care that this wire cannot be shorted to the chassis at any point. This wire will report Zone 5.
H1/10 BROWN (+) siren output Connect this wire to the red wire of the siren. Connect the black wire of the siren to (-) chassis ground, preferably at the same point you connect the control modules black ground wire.
H1/11 RED (+)12V constant power input Before connecting this wire, remove the supplied fuse. Connect to the battery positive terminal or the constant 12V supply to the ignition switch. NOTE: Always use a fuse within 12 inches of the point you obtain (+)12V. Do not use the 15 fuse in the harness for this purpose. This fuse protects the module itself. H1/12 RED/WHITE (-) 200 mA channel 2 output When the system receives the code controlling Channel 2 for longer than 1.5 seconds, the red/white will supply an output as long as the transmission continues. This is often used to operate a trunk/hatch release or other relay-driven function.
IMPORTANT! Never use this wire to drive anything but a relay or a low-current input! The transistorized output can only supply 200 mA of current. Connecting directly to a solenoid, motor, or other high-current device will cause it to fail.
Door Lock Harness (H2) Wire Connection Guide

H2/A H2/B H2/C

___ Green ___ Empty ___ Blue
(-) Lock, (+) Unlock Output Unless Using 451M (-) Unlock, (+) Lock Output
For detailed instructions on wiring the vehicles door locks, please refer to the Door Lock Wiring D Guide (Document No. 1041) provided on the www. directechs.com website or through DirectFax 1-800-999-1FAX (1329). 1

Plug-In Harnesses

Super Bright LED, 2-Pin White Plug
The super bright LED operates at 2V DC. Make sure the LED wires are not shorted to ground as the LED will be damaged. Multiple LEDs can be used, but they must be wired in series. The LED can be top-mounted or flush-mounted. If top-loading the LED with a bezel, the LED fits into a 5 /16-inch mounting hole. If flush-mounting the LED from the back of a panel, drill a mounting hole using a 17/64-inch drill bit. Be sure to check for clearance prior to drilling the mounting hole.
Valet/Program Switch, 2-Pin Blue Plug
The Valet/Program switch should be accessible from the drivers seat. It plugs into the blue port on the side of the unit. Since the system features Valet by using the remote transmitter, the switch can be well hidden. Consider how the switch will be used before choosing a mounting location. Check for rear clearance before drilling a 9/32-inch hole and mounting the switch.The GRAY wire in the two-pin plug may also be used as a (+) Ghost Switch input and can be connected to any (+) switch in the vehicle. (See Feature Descriptions section of this guide.)
Programmer Interface, 3-Pin Black Plug
The black three pin port is provided for either a DEI Bitwriter (P/N 998T) or a personal computer using a PC interface (P/N 996T) to program the unit. When using the Directed Bitwriter or 10
optional PC Interface Module (P/N 996T) it is possible to configure any or all of the programmable functions. The PC Interface Module works with an IBM compatible PC. (The 998T does not require the IBM compatible PC.) For more information please refer to the guide packaged with the programmer.
On-Board Dual Stage Shock Sensor
There is a dual-stage shock sensor inside the control unit. Adjustments are made via the rotary control as indicated in the diagram. Since the shock sensor does not work well when mounted firmly to metal, we do not recommend screwing down the control module. The full trigger of the on-board shock sensor reports Zone 2. See the Table of Zones section of this guide. NOTE: When adjusting the sensor, it must be in the same mounting location that it will be after the installation is completed. Adjusting the sensor and then relocating the module requires readjustment.

Optional Sensor Harness, 4-Pin Connector
The four-pin sensor harness is optional, and is not included with this unit. RED and BLACK wires These wires supply constant 12 volts and ground to the optional sensor. BLUE and ORANGE wires, Zone 4 These wires are multiplex inputs. If a (-) input of less than 0.8 seconds is supplied to either wire, the Warning Zone response will occur. A (-) input of longer than 0.8 seconds to either wire will initiate the triggered sequence and report Zone 4.

Programming Jumper

Light Flash Jumper
This jumper is used to determine the light flash output. In the (+) position, the on-board relay is enabled and the unit will output (+)12V on the WHITE wire, H1/2. In the (-) position, the onboard relay is disabled. The WHITE wire, H1/2, will supply a (-) 200 mA output suitable for driving factory parking light relays. NOTE: For parking light circuits that draw 10 amps or more, the jumper must be switched to a (-) light flash output. P/N 8617 or a standard automotive SPDT relay must be used on the H1/2 light flash output harness wire.

Bypassing Sensor Inputs

There are times when you need to temporarily bypass all sensor inputs to the unit, such as when remote starting the vehicle. Anytime an auxiliary channel output is used, all inputs are bypassed for 5 seconds. During the 5-second bypass period, ground can be supplied to the H1/6 Blue wire without triggering the unit. When the 5 second bypass period ends, if the unit sees ground on the H1/6 Blue wire, all trigger inputs except the door trigger input will remain bypassed until 5 seconds after ground is removed from the BLUE wire. This can be done using the status output of a Directed Electronics remote start unit as shown in the following diagram:
System Features Learn Routine
The System Features Learn Routine dictates how the unit operates. Due to the number of steps, they have been broken up into two menus. It is possible to access and change any of the feature settings using the Valet/Program switch. However, this process can be greatly simplified by using the optional Directed Bitwriter (P/N 998T) or Personal Computer Interface (P/N 996T). Any of the settings can be changed and then assigned to a particular transmitter, up to four, a feature called Owner Recognition. Each time that particular transmitter is used to disarm the system, the assigned feature settings will be recalled. Owner Recognition is only possible when programming the unit via the 996T or the 998T Directed Bitwriter. If using the Directed Bitwriter or PC Interface to program System Features Code Learning, you may lock the unit so that the features cannot be altered via manual programming with the Valet switch. If you later wish to program the system manually, you must unlock the unit using the Directed Bitwriter or PC Interface before you will be able to reprogram the features. If the siren generates one long chirp when attempting to program the unit, this indicates that the unit has been locked and must be unlocked with the Bitwriter or PC Interface before proceeding. 1. Open a door: (The H1/5 GREEN wire or the H1/7 VIOLET wire must be connected.)

Menu #1 - Basic Features

Feature Number 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 1-10 One Chirp Setting Active arming Chirps ON Ignition controlled door locks ON* Active locking only Panic with ignition on 0.8 second door lock pulses Forced passive arming ON Automatic engine disable ON Armed When Driving (AWD) Anti-Code Grabbing ON Two-Chirp Setting Passive arming Chirps OFF Ignition controlled door locks OFF* Passive locking No panic with ignition on 3.5 second door lock pulses Forced passive arming OFF Automatic engine disable OFF Vehicle Recovery System (VRS) Anti-Code Grabbing OFF
Menu #2 - Advanced Features
Feature Number 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 One Chirp Setting Siren 30-second siren duration* False Alarm Control Technology ON Progressive door trigger Valet switch input: 1 pulse Door trigger error chirp ON Ignition-controlled domelight ON Single unlock pulse Channel 3: Validity Two-Chirp Setting Horn honk 60-second siren duration* False Alarm Control Technology OFF Instant door trigger Valet switch input: 2-5 pulses Door trigger error chirp OFF Ignition-controlled domelight OFF Double unlock pulse Channel 3: latched/latched, reset with ignition/30-second timed/ second unlock output

Feature Descriptions

The features of the system are described below. Features that have additional settings that can be selected only when programming with the PC interface or Bitwriter are indicated by the following icon:
1-1 ACTIVE/PASSIVE ARMING: When active arming is selected, the system will only arm when the transmitter is used. When set to passive arming, the system will arm automatically 30 seconds after the last door is closed. To alert the driver of passive arming, the siren will chirp 20 seconds after the door is closed. This provides the driver with an audible warning prior to the system actually arming. At the 30 second mark, the system will arm but the siren will not chirp. 1-2 CHIRPS ON/OFF: This controls the chirps that confirm the arming and disarming of the system. 1-3 IGNITION CONTROLLED DOOR LOCKS ON/OFF: When turned on, the doors will lock three seconds after the ignition is turned on and unlock when the ignition is turned off. The TechSoft Programmer (P/N 996T) or the Bitwriter (P/N 998T) will display separate steps for ignition lock and ignition unlock. They can be programmed on or off independently. 1-4 ACTIVE/PASSIVE LOCKING: If passive arming is selected in Feature 1-1, then the system can be programmed to either lock the doors when passive arming occurs, or only lock the doors when 16

the system is armed with the transmitter. Active locking means the system will not lock the doors when it passively arms. Passive locking means that the system will lock the doors when it passively arms. NOTE: Remember, when passive arming is selected, the unit will chirp 20 seconds after the last door is closed. The system does not actually arm or lock the doors until 30 seconds after the door has been closed. 1-5 PANIC WITH IGNITION ON: This feature controls whether or not the Panic Mode is available with the ignition on. In some states, there are laws prohibiting a siren from sounding in a moving vehicle. This feature allows the system to be compliant with these regulations. 1-6 DOOR LOCK PULSE DURATION: Some European vehicles, such as Mercedes-Benz and Audi, require longer lock and unlock pulses to operate the vacuum pump. Programming the system to provide 3.5 second pulses, will accommodate the door lock interface in these vehicles. The default setting is 0.8 second door lock pulses. 1-7 FORCED PASSIVE ARMING ON/OFF: To use this feature, passive arming must be selected in Feature 1-1. When turned on, forced passive arming will ensure that the system will passively arm, even if a zone is left open or invalid. Forced passive arming occurs one hour after the ignition is turned off. 1-8 AUTOMATIC ENGINE DISABLE (AED) ON/OFF: AED is a full-time, passive starter disable that functions independent of the security system. When turned on the H1/1 ORANGE groundwhen-armed output will activate 30 seconds after the ignition is turned off. The LED will flash at half its normal rate when the ignition is turned off to indicate that AED is active and will interrupt the starter in 30 seconds. AED does not occur in Valet mode and can be bypassed using the emergency override procedure. The transmitter can be used to disarm AED; however, the system must be armed and then disarmed with the transmitter, in order to disarm AED. 1-9 ARMED WHILE DRIVING/VEHICLE RECOVERY SYSTEM: In the default setting (Armed While Driving), the system can be armed with the ignition on. When armed, the ground-whenarmed is not active and the sensors are bypassed. The door triggers will remain active. If programmed to the Vehicle Recovery System (VRS) setting, VRS will be activated. 1-10 ANTI-CODE GRABBING ON/OFF: The system uses a mathematical formula to change its code each time the transmitter and receiver communicate. This makes the group of bits or word from the transmitter very long. The longer the word is, the easier it is to block its transmission to the unit. Disabling this feature lets the receiver ignore the Anti-Code Grabbing part of the transmitted word. As a result, the unit may have better range with the Anti-Code Grabbing feature off.
2-1 SIREN/HORN HONK: The system can be programmed to output pulses instead of a continuous output when the system is triggered. This is useful to honk the factory horn in applications where a siren is undesirable. Remember that the unit is only capable of supplying 1 amp of current. A relay will be required to interface with most factory horn systems. 2-2 SIREN DURATION 30/60 SECONDS: It is possible to program the unit to sound for 30 or 60 seconds during the triggered sequence. Some states have laws regulating how long a security system can sound. When using the TechSoft Programmer or Bitwriter, the siren can be programmed to sound for any length of time ranging from 1 to 180 seconds. Use the right and left arrows or the + and - keys on your keyboard to change the siren duration in 1 second intervals. Holding down the key will rapidly increase or decrease the setting. The desired siren duration can also be directly entered by using the number keys on your computer's keyboard. When using the Bitwriter, pressing the SELECT button will adjust the siren duration. 2-3 FALSE ALARM CONTROL TECHNOLOGY (FACT) ON/OFF: FACT stops repeated triggering of the same zone. If one zone is triggered three times in one hour, that zone is bypassed for one hour, starting from the time of the third trigger. During that hour, if the system detects a trigger on that zone again, the system resets the one hour timer. If one hour passes and the zone has not triggered again, the zone is activated and can trigger the system again. FACT monitors sensor inputs and door triggers, but does not bypass the ignition trigger at any time. If FACT is turned off, the system will respond to repeated triggers on the sensor inputs and will do so indefinitely. Some states have laws regulating how many times a security system can trigger before it is considered a nuisance and the vehicle is towed away. 2-4 PROGRESSIVE DOOR TRIGGER ON/OFF: The security system responds to a door trigger input with a progressive response. When the door is opened with the system armed, the siren will chirp 10 times prior to the full triggered sequence. The door trigger is still treated as an instant trigger and closing the door quickly will not prevent a full triggered sequence from occurring. If the progressive door trigger is programmed off, the full siren output will occur the moment the door is opened. 2-5 VALET PULSE COUNT 1-5 PULSES: The security system can be programmed to count the number presses of the valet switch before disarming the security system or VRS. The factory default setting is one pulse. The unit can be set for 2 to 5 pulses using the two-chirp setting to select the pulse count. Ghost Switch option: For added security, the GRAY wire on the two-pin Valet/Program switch can be connected to any switch in the vehicle that provides a positive (+) momentary pulse. 18

2-6 DOOR TRIGGER ERROR CHIRP ON/OFF: With the door trigger error chirp programmed off, the system will not report an invalid zone on arming when the door trigger wire is active. This eliminates the extra chirps that occur when interfacing with vehicles that have exceptionally long dome light delay circuits. 2-7 IGNITION-CONTROLLED DOMELIGHT SUPERVISION ON/OFF: If turned on, the system will turn on the domelight for 30 seconds when the ignition is turned off. The optional domelight supervision feature must be installed. 2-8 DOUBLE PULSE UNLOCK ON/OFF: Some vehicles require two pulses on a single wire to unlock the doors. When the double pulse unlock feature is turned on, the BLUE H2/C wire will supply two negative pulses instead of a single pulse. At the same time, the GREEN H2/A wire will supply two positive pulses instead of a single pulse. This makes it possible to directly interface with double pulse vehicles without any extra parts. 2-9 CHANNEL 3 VALIDITY/LATCHED/LATCHED RESET WITH IGNITION/30 SECOND TIMED/SECOND UNLOCK OUTPUT: Channel 3 can be programmed for any of these output configurations. The unit is set to the default validity output. To change the configuration, use the two-chirp setting to toggle to the different configurations. (Refer to the H1/3 WHITE/BLUE wire description in the Primary Harness (H1) Wire Connection Guide section of this guide.)
Transmitter/Receiver Learn Routine
The system comes with two transmitters that have been taught to the receiver. The receiver can store up to four different transmitter codes in memory. Use the following learn routine to add transmitters to the system or to change button assignments if desired. If the Directed Bitwriter or PC Interface has previously been used to program the system, the unit may have been locked, so that the features and channels cannot be altered via manual programming with the Valet switch. If the siren generates one long chirp when attempting to program the transmitter/receiver, this indicates that the unit is locked. You must unlock it with the Bitwriter or PC Interface before you will be able to manually program the transmitter/receiver. NOTE: The transmitters cannot be programmed using the Bitwriter or the PC Interface. 1. Open a door: (The H1/5 GREEN wire or H1/7 VIOLET wire must be connected.)
2. Key: Turn the ignition on. (The YELLOW wire, H1/9 must be connected.)
3. Select the receiver channel: Press and release the Valet/Program switch the number of times necessary to access the desired channel. Press and HOLD the Valet/Program switch once more. The siren will chirp and the LED will blink the number of times corresponding to the channel that has been accessed.

Channel Number 9

Function Arm/Disarm/Panic Silent Mode/Remote Valet/Trunk Release Remote Start or other accessories Arm only (only available with Radar Master remote)1 Disarm only (only available with Radar Master remote)1 Panic only (only available with Radar Master remote)1 Auto-learn 3-Button Transmitter Configuration2 Auto-learn 4-Button Transmitter Configuration2 Zap (delete all transmitters)3

Wire Color RED/WHITE WHITE/BLUE
NOTE: Channels 4-6 are only available when using an optional Radar Master remote. NOTE: For Auto Learn Configurations, see Transmitter Configurations section of this guide. 3 NOTE: See Channel 9 description in this section. 4. Press the transmitter button: While HOLDING the Valet/Program switch, press the button from the transmitter that you wish to assign to the selected channel. The unit will chirp once indicating that the channel has been entered. 5. Press the same transmitter button again: While still HOLDING the Valet/Program switch, press the same transmitter button that you just programmed. The siren will chirp twice to confirm that the desired channel has been successfully programmed to the transmitter button. If this step is not performed, the channel will not be programmed to the transmitter button. It is not possible to teach a transmitter button to the system more than once. 6. Release: Once the code is learned, the Valet/Program button can be released.
Channels 4-6 (available only when using a Radar Master remote) Channels 4 through 6 are used to assign the arm, disarm and panic functions to separate buttons on the remote control. These channels are only available when using an optional Radar Master remote. (See Transmitter Configurations section of this guide.) Teaching a transmitter button to Channel 4 erases all previous programming from the transmitters memory. Similarly, if the transmitter is set up to use the separate arm, disarm and panic channels and a button from that transmitter is programmed to Channel 1, the transmitters memory will be erased, and the system will only recognize the button that was programmed to Channel 1. Channel 9 If any transmitter button from a known transmitter is programmed to Channel 9, all transmitters will be erased from memory and will revert to the default feature settings (see the Features Menu section of this guide). This is useful in cases where the one of the customer's transmitters is lost or stolen. This will erase any lost or stolen transmitters from the system's memory. It can also be used to start from scratch if the transmitter buttons were programmed incorrectly. To advance from one channel to another: You can advance from programming one channel to another by releasing the Valet/Program switch and tapping it to advance channels and then HOLDING it. For instance: You have programmed Channel 1 and you want to program Channel 2. Release the Valet/Program switch. Press it one time and release it to advance from Channel 1 to Channel 2. Now, press and HOLD the Valet/Program switch. The LED will flash two times and the siren will chirp twice (if connected). As before, do not release it. To exit the learn routine: One long chirp indicates that Learn Routine has been exited. Learn Routine will be exited if any of the following occurs:

Ignition is turned off. Door is closed. Valet/Program button is pressed too many times. More than 15 seconds elapse between steps.
Transmitter Configurations
Using the Auto Learn functions in the Transmitter/Receiver Learn Routine, the transmitters can be programmed either with the 3-button (standard) configuration or 4-button configuration, which requires an optional 4-button transmitter.
3-Button Transmitter Configuration
This configuration can be programmed to a 3-button transmitter using Channel 7 of the Transmitter/Receiver Learn Routine. The transmitter buttons are assigned to the following functions:.operates..Arm/Disarm/Panic..operates..Channel 2..operates..Channel 3
4-Button Transmitter Configuration
This system has a 4-button transmitter configuration that can be used when upgrading to an optional 4-button remote. This configuration can be programmed to a 4-button transmitter using Channel 8 of the Transmitter/Receiver Learn Routine. In the 4-button transmitter configuration, the buttons are assigned to the following functions:.operates..Arm/Disarm/Panic..operates..Channel 2..operates..Channel 3.is..Unassigned
Optional Radar Master Transmitter
Separate transmitter button arming/disarming/panic (Channels 4-6, see channel chart in Transmitter/Receiver Learn Routine section) can only be utilized when upgrading to an optional Radar Master transmitter. When using a Radar Master transmitter with this system, Channels 4-6 may be programmed to the transmitter in a variety of configurations.
Multi-Level Security Arming
Multi-Level Security Arming is only available when using an optional Radar Master transmitter that has been configured with separate transmitter buttons for arming and disarming. Multi-Level Security Arming allows you to select which of the system's inputs or sensors will be active or bypassed at the time that the system is armed. (See Table of Zones section.) Pressing the arm button again within five seconds of arming the system will activate Multi-Level Security Arming. Each time the arm button is pressed again, a different security level is selected. The different security levels can be selected as follows:
Pressing the arm button once: The siren chirps once. The system is armed. Pressing the arm button twice within five seconds: The siren chirps twice followed by a long chirp. Zone Two is now bypassed. Pressing the arm button a third time within five seconds: The siren chirps three times followed by a long chirp. Zone Four is now bypassed. Pressing the arm button a fourth time within five seconds: The siren chirps four times followed by a long chirp. Zones Two and Four are now bypassed. Pressing the arm button a fifth time within five seconds: The siren chirps five times followed by a long chirp. All input zones, except the ignition, are now bypassed.

NOTE: Multi-Level Security Arming only applies to a single arming cycle. Once the system is disarmed and then re-armed, all the zones will be active again.

Smart Power Up II

The Smart Power Up II feature ensures that when the security system is powered back up after power has been disconnected, the system will resume the same state it was in before power was lost. For example, if power is disconnected during a full trigger sequence, the system will still be in the full trigger sequence when power is reconnected to the unit. If power is disconnected while the unit is disarmed, it will still be disarmed when power is restored. This also applies to the VRS sequence. If the unit loses power at any time during the VRS sequence, it will automatically resume the VRS full trigger sequence when the unit is powered back up.

Table of Zones

When using the diagnostic functions, use the Table of Zones to see which input has triggered the system. It is also helpful in deciding which input to use when connecting optional sensors and switches. NOTE: The Warning Zone response does not report on the LED.
Zone No. Trigger type Instant On-board shock sensor Two-stage, progresses from warning to full alarm Multiplexed Input Input description H1/6 BLUE wire. Connect to optional hood/trunk pins. Heavy impact detected by the on-board shock sensor. Door switch circuit. H1/5 GREEN or H1/7 VIOLET. BLUE and ORANGE wires of optional sensor plug. Inputs shorter than 0.8 seconds will trigger a Warning Zone response, while inputs longer than 0.8 seconds will instantly trigger a full alarm sequence and report Zone 4. Ignition input. H1/9 YELLOW.
Two-stage (similar to doors)

Long Term Event History

The system stores the last two full triggers in memory. These are not erasable. Each time the unit detects a full trigger, the older of the two triggers in memory will be replaced by the new trigger. To access long term event history: 1. With the ignition off, press and HOLD the Valet/Program switch.

Turn on the ignition.

3. Release the Valet/Program switch.
4. Press and release the Valet/Program switch within 5 seconds. The LED will flash in groups indicating the last two zones that triggered the unit. The LED will flash for one minute or until the ignition is turned off. NOTE: The Warning Zone triggers are not stored to memory and will not be reported.

Optional Vehicle Recovery System (VRS)
No additional parts are required to add the optional VRS feature. However for the VRS feature to be effective, the 8618 Starter Kill Relay must be installed. The VRS feature can be activated with the remote transmitter and deactivated with the Valet switch. If the VRS option is selected it is recommended to program the Valet switch to respond to more than one pulse for maximum security. (See System Features Learn Routine section.) NOTE: For a detailed explanation to the VRS triggered sequence, refer to the Vehicle Recovery System section of the Owner's Guide.

To Arm the VRS

1. Turn the ignition to the ON position.

2. Press

on the transmitter for 1 second.
The parking light will flash once and the siren will chirp once to confirm that the VRS system is armed and will enter the trigger sequence next time a door is opened and then closed.

To Disarm the VRS

To disarm the VRS (before the siren begins chirping): 1. Make sure that the ignition is OFF before beginning the disarm procedure.
Turn the ignition to the ON position.
3. Press on the transmitter for 1 second. The parking lights will flash twice and the siren will chirp twice to confirm that the VRS is disarmed.
To disarm VRS (after the siren has begun chirping): 1. Make sure that the ignition is OFF before beginning the disarm procedure.
2. Turn the ignition to the ON position.
3. Press and release the Valet/Program switch the selected number of times programmed in Feature 2-5. (See System Features Learn Routine section of this guide.) NOTE: After the VRS has entered the chirp sequence, the transmitter will not disarm the VRS. The Valet switch must be used.
False Alarm Control Technology (FACT)
FACT requires that you change the way you test the system, as FACT will bypass an input zone for 60 minutes. If the system detects the same zone triggering three times AND the triggers are spaced less than an hour apart, the system will bypass that input zone for 60 minutes. If that zone does not attempt to trigger the system during the 60-minute bypass period, the zones monitoring will begin again at the end of the hour. If it does attempt to trigger while bypassed, the 60-minute bypass starts over again. Disarming and rearming the system does not reset FACT. The only way to reset FACT is for the 60 minutes to pass, without a trigger, or for the ignition to be turned on. This allows the system to be repeatedly triggered, disarmed and rearmed, while still allowing FACT to bypass a faulty zone. When disarming the system, 5 chirps indicate FACT is activated. The LED will report the zone that has been bypassed. (See Table of Zones section of this guide.) 26

Troubleshooting

Starter kill does not work. Is the correct starter wire being interrupted? If the car starts when the starter kill relay is completely disconnected, the wrong starter wire has been cut and interrupted. Yellow wire is not connected to true ignition. It is connected to an accessory circuit. Shock sensor does not trigger the alarm. Has the FACT system been triggered? If so, you will hear five chirps when disarming. To check this, turn the ignition key on and off to clear the FACT from memory, and then retest the shock sensor. For a detailed description of FACT, see False Alarm Control Technology section of this guide. Door input does not immediately trigger full alarm. Instead, I hear chirps for the first three seconds. Thats how the progressive two-stage door input works! This is the instant response feature of this system. Even if the door is closed immediately, the system provides an instant trigger by chirping, and then progressing to a constant siren. Closing the door triggers the system, but opening the door does not. Have you correctly identified the type of door switch system? This happens often when the wrong door input has been used. System will not passively arm until it is remotely armed and then disarmed. Are the door inputs connected? Is a blue wire connected to the door trigger wire in the vehicle? Either the green H1/5 or the violet H1/7 should be used instead. Door input does not respond with the progressive trigger, but with immediate full alarm. What zone does the LED indicate? If the LED indicates that the impact sensor caused the trigger, the sensor may be detecting the door opening. Reducing the sensitivity or relocating the sensor can often solve this problem. If the LED indicates that the door caused the trigger, you may have programmed the progressive door trigger off. (See Feature 2-4 in the Feature Descriptions section of this guide.) The Valet switch does not work. Is it plugged into the correct socket? Check the System Features Learn Routine for the programmed Valet pulse count. Status LED does not work. Make sure that it is plugged in. (See Plug-In Harnesses section of this guide.) Is the LED plugged into the correct socket?

Georgian Mathematical Journal Volume 12 (2005), Number 1, 1525
ON SOME MATRIX CLIFFORD ALGEBRAS
TAMAZ CHANTLADZE, NODAR KANDELAKI, AND DOUGLAS UGULAVA
Abstract. A sequence of matrices U1 , U2 ,. , Um is constructed, which satises the conditions Ui Uj = Uj Ui (i = j), Ui2 = I. These matrices are used to construct representations of a Cliord algebra for special quadratic forms. 2000 Mathematics Subject Classication: 15A66. Key words and phrases: Cliord algebra, quadratic form, matrix algebra, associative ring.
1. Statement of the Problem In this paper we propose one more construction of Cliord algebras. Though several explicit representations are available in the literature on Cliord algebras (see, e.g., [1][6]), we think that our construction is also of interest and can be used, say, for obtaining matrix representations of innite dimensional Cliord algebras. Let R be an associative ring with unity (1) and 1 R. 2 Denition 1.1 ([2], Ch. 11). A pair (E, f ) consisting of some free R-module E and a bilinear functional f on this module will be called a quadratic form over the ring R. For a given quadratic form (E, f ) we denote by A the set of all pairs (A, h), where A is an associative R-algebra with unity, and h is a homomorphism of R-modules from E into A such that h2 (x) = f (x, x) 1 for all x E. Denition 1.2 ([2], Ch. 11; [3]). We call a Cliord algebra of the quadratic form (E, f ) a pair (C(f ), g) A such that for any pair (A, h) A there exists a uniquely dened homomorphism w of R-algebras from C(f ) into A for which the diagram E DD

DD D g DDD !

/ =A zz z zz zz w z
C(f ) is commutative, i.e., a Cliord algebra is an initial object in the category A if as morphisms between objects of this category we take homomorphisms between algebras that constitute diagrams similar to the above one. Let (E, f ) be a given quadratic form satisfying the following conditions: 1) E is the unital module over R (i.e., 1 x = x for all x from E);
ISSN 1072-947X / $8.00 / c Heldermann Verlag www.heldermann.de
T. CHANTLADZE, N. KANDELAKI, AND D. UGULAVA
2) E has a basis e1 , e2 ,. , em 1 f (ei , ei ) = 0 1
such that f (ei , ej ) = 0 for all i = j and for 1 i t, for t < i t + q, for t + q < i m = t + q + k, (1.1)
where k, t, q are non-negative integers. As is known (see, for example, [1][3]), a Cliord algebra of the quadratic form (E, t) exists, its dimension is 2m , and its generators U1 , U2 ,. , Um satisfy the so-called Cliord relations Ui2 = f (ei , ei ) 1, Ui Uj = Uj Ui , 1 i, j m. For simplicity we will give a detailed description of the properties of some matrices which we need for our further discussion and which occur in the aforementioned well-known constructions. 2. Some Definitions and Facts from Matrix Theory Let R be the same as above, and Mat R(n) be the algebra of all nn matrices with coecients from R. Denition 2.1. The set of aii , i = 1, 2,. , n, is called the diagonal of the matrix A = (aij ). Denition 2.2. The set of aij such that i + j = n + 1, i, j = 1, 2,. , n, is called the contradiagonal of the same matrix. Denition 2.3. We call a nonzero matrix diagonal (contradiagonal) if all nonzero elements lie on the diagonal (contradiagonal). Now let A = (aij ) Mat R(n) and B = (bij ) Mat R(m). Denition 2.4. We call the direct product of the matrices A and B (= AB) the matrix C = (cij ) Mat R(n + m), along whose diagonal there are the matrix A in the left-hand upper corner and the matrix B in the right-hand lower corner; more exactly, cij = aij when i, j = 1, 2,. , n and c(n+t)(n+h) = bth , where t, h = 1, 2,. , m; the rest are zeros: C= A 0. 0 B
Denition 2.5. We call the skew product of the same matrices A and B (= A#B) the matrix D = (dij ) Mat R(n + m), where di(m+j) = aij , i, j = 1, 2,. , n, and d(n+i)j = bij , i, j = 1, 2,. , m; the rest are zeros: D= 0 A. B 0
Let Ai Mat R(n), i = 1, 2,. , k; then their direct products and skew products are dened by means of induction and we respectively denote by Ai
the direct product, and by # Ai the skew product (we omit the product limits when the context clearly implies them). For r R, r Ai = (r Ai ) and r #Ai = #(r Ai ) (analogously from the right). Remark 2.1. Let A Mat R(n) and B Mat R(m), and let m divide n, i.e., n = km for some natural k. Then A can be treated as a matrix of k k matrices, where the dimension of each matrix cell is m m, i.e., Mat R(n) = Mat(Mat R(m))(k); here the ring of coecients is Mat R(m). Under the product B A (or A B) we naturally understand the product of the number B by the matrix (or vice versa), i.e., by the corresponding matrix cell in A, from the right or the left, respectively. One may come to the same result by taking a kmultiple direct product B by itself (= B) and multiplying by A (or vice versa) in Mat R(n). The B can also be interpreted as a product of the coecient B by the unit matrix In Mat R(n). Moreover, the mapping A A = A In gives a homomorphism of algebras Mat R(m) is embedded into Mat R(n). Let B Mat R(m), Ai Mat R(ni ), i = 1, 2,. , h, and let m divide all ni. Then the following equalities are valid: B Ai = (B Ai ), B #Ai = #(B Ai ), (Ai ) B = (Ai B), (#Ai ) B = #(Ai B). 3. Construction of Clifford Matrix Algebras Using the notation we have introduced above, we will consider the matrix algebras Mk = Mat R(2k ), k = 1, 2,. , n, where n is a xed natural number. For all 0 k n, we dene Vk Mk = Mat R(2k ), Vk = (vij ) and vij = for i + j = 2k + 1, in other cases, (2.1) (2.2)

i.e., Vk is a contradiagonal matrix from Mk with units on the contradiagonal. For any 0 i k, Vk can be written as a 2ki -multiple skew product Vi Mi : Vk = #Vi (3.1) i.e., the contradiagonal of the matrix Vk is partitioned into 2ki nonintersecting parts, each of which contains 2i units and represents the contradiagonal Vi. In particular Vk = Vk1 #Vk1 , i.e., Vk = 0 VkVk1 Vk1. 0 Vk1. 0
Moreover, assuming that Wk = (Vk1 )#Vk1 , we have Wk =
2 Proposition 3.1. For any k 1: 1) Vk2 = Ik ; 2) Wk = Ik ; 3) if i j, then Wi Vj = Vj Wi , and if i > j, then Wi Vj = Vj Wi.
Here and in what follows Ik is a unit matrix in Mk = Mat R(2k ) for any natural k. Proof. 1) (Vk2 )ij = vih vhj. This sum for xed i, j 2k is dierent from zero if and only if i + h = 2k + 1 and j + h = 2k + 1. This means that (Vk2 )ij = ij and therefore Vk2 = Ik. VkVkVk2) Wk = = 2 Vk0 Vk1 VkIk= = Ik. 0 Ik1 3) Assume rst that i = j. Then Wk Vk = = Vk Wk = = 0 Vk1 Vk0 Vk1 Vk=

2 VkVk1

Ik, 0 IkVk1 Vk0 Vk1 Vk=

Ik0 Ik1

and therefore 3) is fullled for i = j. If i < j, we have Vj = #Vi (see (3.1)). Thus Wi Vj = Wi (#Vi ) = #Wi Vi = #Vi Wi = (#Vi ) Wi = Vj Wi. If i > j, then Wi = Vi1 #Vi1 , Vi1 = #Vj (the right-hand side is a 2i1j multiple skew product of the matrix Vj by itself). Therefore Wi Vj = [(#Vj )#(#Vj )] Vj = (#Vj Vj )#(#Vj Vj ) = Vj ((#Vj )#(#Vj )) = Vj Wi and Proposition 3.1 is completely proved. Proposition 3.2. Wi Wj = Wj Wi when i = j. Proof. Let i < j. By denition, Wj = (Vj1 )#Vj1. Since i j 1, by virtue of (3.2) Wj = (#Vi )#(#Vi ) (where #Vi is a 2j1i -multiple skew product Vi by itself). Then Wi Wj = Wi [(#Vi )#(#Vi )] = (#Wi Vi )#(#Wi Vi ) = (# Vi Wi )#(# Vi Wi ) = [(#Vi )#(#Vi )] Wi = Wj Wi (see (2.1) and (2.2) and also 3) from Proposition 3.1). Let us consider the matrices U1 , U2 ,. , Un from Mn dened as follows Ui = Wi In , i = 1, 2,. , n, i.e., Ui = Wi (see Remark 2.1). Lemma 3.1. 1) Ui2 = In ; 2) Ui Uj = Uj Ui when i = j, i, j = 1, 2,. , n.
Proof. 1) Ui2 = (Wi In )2 = (Wi In )(Wi In ) = Wi Wi In = Wi2 In = Ii In = In. 2) Ui Uj = (Wi In )(Wj In ) = Wi Wj In = Wj Wi In = Wj Ui = Wj In Ui = Uj Ui (see Remark 2.1 and Proposition 3.2). Let us consider the smallest subalgebra n Mn containing all Ui. By virtue of Lemma 3.1 this subalgebra coincides with the R-module, which is generated by elements Ui1 Ui2 Uik with indices 1 i1 < i2 < < ik n (an empty product is, as usual, equated to the unit of the algebra In ). Let us consider the sets of the generators Bk = Ui1 Ui2 Uih |1 i1 < i2 < < ih k , It is easy to see that Bk+1 = Bk (Bk Uk+1 ). (3.2) Lemma 3.2. For any k, 1 k n, and any 1 i 2k there exists a unique (i) matrix Ai = (ath ) Bk such that a1h = a2n l =

(1) (1)

k = 1, 2,. , n.

(1)h+1 , 0, 1, 0,

for h = i, for h = i,

(3.3) (3.4)

for l = 2n i + 1, for l = 2n i + 1.
Proof. To prove the lemma we use the method of induction. If k = 1, then B1 = {In , U1 }, 1 i 2, and the matrices A1 = In , A2 = U1 satisfy conditions (3.3) and (3.4). Supposing now that the lemma holds for Bk , let us prove it for Bk+1. For all i, 1 i 2k , the respective matrices Ai Bk satisfy, by induction, conditions (3.3) and (3.4). Let now 1 i 2k+1. Since, by (3.2), Bk Bk+1 , we are to consider only the case where 2k < i 2k+1. For given i, 2k < i 2k+1 , we dene Ai = Aj Uk+1 , where j is dened from the condition j + i = 2k+1 + 1. Since 2k < i 2k+1 , we have 1 j 2k. Then (i) (j) (i) a1i = a1j = (1)j+1 = (1)i = (1)i+1 , a1h = 0 if h = i, equalities (3.3) (j) are valid. Indeed, by induction, a1h = (1)h+1 if h = j and is equal to zero otherwise. Therefore the rst row of the matrix Aj multiplied by the respective column of the matrix Uk+1 gives a nonzero element provided that in this column a nonzero element is in the j-th position. By denition, we have Uk+1 = Wk+i In and the nonzero elements of this matrix lie, within the desired limits, on the contradiagonal of the rst component of the direct product Wk+1 , i.e., are dened from the equality i + j = 2k+1 + 1, while we need only the upper half of the contradiagonal, where there are only (1) s. By induction, in the matrix Aj the last row has 1 in the (2n j + 1)-th position, while in the matrix Uk+1 the row with the same number has 1 in the column whose number is 2n 2k+1 + j. After multiplying these matrices, the last row will have unit in the column whose number is l = 2n 2k+1 + j. Taking into account that j = 2k+1 + 1 i, we obtain l = 2n i + 1. We thereby obtain equalities (3.4) and Lemma 3.2 is proved.
Remark. Applying the method of induction, we make certain that in each row and each column of matrices from the set Bk there exists a unique nonzero element. Lemma 3.3. Any Bk is a set of vectors that are linearly independent over R. Proof. Instead of Ui and their product we will deal, using Lemma 3.2, with matrices Ai. Thus we are to prove linear independence of Ai , 1 i 2k , over R. Let

2k i=1

ai Ai = 0 and consider the rst row of the total matrix. By condition, the
latter matrix must consist of zeros only: (a, a,. , a2k 2k ) = (0, 0,. , 0), i.e., ai i = 0 (1 i 2k , since the number of strictly ordered sequences in Bk , the empty one inclusive, is equal to 2k ). Since all i = 1, it follows that all ai are equal to zero. Let k and t be non-negative numbers such that k + t = m, and consider the following matrices from Mat R(2m ): D1 = W1 Vt Im , D2 = W2 Vt Im ,. , Dt = Wt Vt Im , Dt+1 = Wt+1 Im ,. , Dm = Wm Im. (3.5)
It is assumed that for t = 0 this system coincides with the system Di = Ui , 1 i m, while for k = 0 with the system Di = Wi Vi Im , 1 i m. By virtue of Propositions 3.1 and 3.2 we make certain that 2 a) if 1 i t, then Di = Wi Vt Wi Vt Im = Wi2 Vt2 Im = Im ; 2 b) if t < i m, then Di = Ui2 = Im ; c) if 1 i, j m, i = j, then Di Dj = Dj Di. Indeed, if 1 i, j t, then Di Dj = Wi Vt Wj Vt Im = Wi Wj Vt2 Im = Wj Vt Wi Vt Im = Dj Di ; if t < i, j m, then (Lemma 3.1) Di Dj = Ui Uj = Uj Ui = Dj Di ; if 1 i t, t < j m, then Di Dj = Wi Vt Wj Im = Wj Wi Vt Im = Dj Di. Thus the following lemma is valid. Lemma 3.4. If m = t + k, then the matrices D1 , D2 ,. , Dm dened by (3.7) satisfy the conditions Di = Im when 1 i t; Di = Im when 1 t + 1 i m, and Di Dj = Dj Di when i = j, i, j = 1, 2,. , m. Lemma 3.5. All ordered products Di1 Di2 Dih for all indices 1 ii < i2 < < ih m (the empty product is Im ) are linearly independent over R. Proof. If t = 0, then Lemma 3.5 coincides with Lemma 3.3 and therefore below it is assumed that t 1. Consider the following sets from Mat R(2m ): T1 = {Im ; Wi1 Wi2 Wih Im }; T2 = {Wi1 Wi2 Wih Vt Im } where the products are considered with respect to possible ordered products of indices 1 i1 < i2 < < ih m, where the number of indices not exceeding t is an even number or zero. All ordered products Di1 Di2 Dih are, up to a sign, the elements of the union T1 T2. To prove the lemma it is sucient to

show that all elements of this union are linearly independent over R. Let us establish a one-to-one correspondence between T1 and T2 as follows: Im Wt Vt Im = Dt , Wi1 Wil Wil+1 Wih Im Wi1 Wil Wt Vt Wil+1 Wih Im ; Wi1 Wil Wt Wil+1 Wih Im Wi1 Wil Vt Wil+1 Wih Im , if 1 i1 < < il < t < il+1 < < ih m. For t = 1 there is no product in T1 that contains W1 as a multiplier, and to an element Wi1 Wih Im T1 , where 1 < i1 < < ih m, we assign the element W1 V1 Wi1. Wih Im T2. If t = m and 1 i1 < < ih < m, then Wi1 Wih Im Wi1 Wih Wm Vm Im , Wi1 Wih Wm Im Wi1 Wih Vm , where the matrices on the right-hand sides belong to T2 , since, by Proposition 3.1, Vt can be placed in front of Im. For 1 i1 < < il < t, order of the matrix Wi1 Wil does not exceed It2t1 , and Wt Vt =. Hence taking into account Lemmata 3.1 and 0 It1 3.2 we conclude that if for 1 i1 < < il < t the unique nonzero element (i) of the rst row in Wi1 Wil Im is a1i = (1)i+1 , then, after multiplying this product by Wt Vt Im , we obtain the matrix, where the nonzero element of the (i) rst row will again occur only in the i-th column and be equal to (1)i = aii. In both matrices, the unique nonzero elements of their last rows are equal to 1 and contained in the column whose number is 2m i + 1. Thus the last rows of the matrices Wi1 Wil Im and Wi1 Wil Wt Vt Im contain units in one and the same columns, while the other elements are zeros. In the rst rows, the nonzero elements (namely, 1 or (-1)) occur only once in one and the same column and dier in sign. In view of the Remark to Lemma 3.2, we make certain that the multiplication of these matrices from the right by Wil+1 Wih gives the matrices, where in the rst and the last row the nonzero elements occur in one and the same place. Note that in the rst row these elements dier in the sign, and in the last row they are equal to 1. The numbers of columns of nonzero elements in the rst and the last row are dened by Lemma 3.2 and dierent pairs have dierent numbers. A similar argument is also applicable to other pairs participating in the correspondence between the sets T1 and T2 described above and thus we come to the following conclusion: (1) (2) (1) (1) If A1 = (aij ) T1 A2 = (aij ) T2 , a1l = 0 (a1l is equal to 1 or to (-1)), (2) (1) (1) (1) (2) a1s = 0, s = l, a2m s = 1, a2m r = 0, r = s, then a1l = a1 , a1s = 0, s = l, 1l (2) (2) a2m s = 1, a2m r = 0, r = s. It is obvious that A1 and A2 are given by the unique matrices from T1 and T2 , respectively, where nonzero elements occur in the aforementioned columns in the rst (and therefore in the last) rows.

Let us now consider an arbitrary linear combination of all elements of the sets T1 and T2 and group together all terms containing matrices that correspond to (1) (2) each other. Let us choose a pair A1 = (aij ) T1 and A2 = (aij ) T2 , where (1) (1) (1) (2) (2) a1l = 0, a1l = 1, a2m s = a2m s = 1, a1l = 1. In their linear combination, the element of the rst row and the l-th column has the form , while the element of the last row and the s-th column has the form +. Other terms of the considered linear combination of T1 and T2 have zeros in the aforementioned positions and therefore the equality of this linear combination to zero means 1 that = 0 and + = 0. Since 2 R, the equalities 2 = 0 and 2 = 0 imply that = = 0, which completes the proof of Lemma 3.5. Assume now that n = t + 2q + k, where t, q, k are non-negative integers and consider the set of matrices D1 , D2 ,. , Dn from Mat R(2n ), which is dened according to (3.5), where m is replaced by the number n, and t by the number t + q. By Lemma 3.5, the set En of all ordered products Di1 Di2 Dih , 1 i1 < i2 < < ih n, is actually the set of linearly independent vectors. Let m = t+q +k be a new set of linearly independent matrices C1 , C2 ,. , Cm from Mat R(2n ), which is dened as follows: Di = Wi Vt+q In , for 1 i t, Ci = Di + Dq+i = Wi Vt+q In + Wq+i In , for t < i t + q, (3.6) D = W I , for t + q < i t + q + k = m. q+i q+i n We can immediately verify that 1, for 2 Ci = 0, for 1, for Ci Cj = Cj Ci , 1 i t, t < i t + q, t + q < i t + q + k = m, 1 i, j m, i = j.
Let us construct the algebra C m with generators C1 , C2 ,. , Cm by using usual matrix multiplication. To each ordered product Ci1 Ci2 Cih we put into correspondence, in En , some subset as follows: 1) if among the indices there is no index ik such that t < ik t + q, then to such a product we assign this product itself; 2) if however among the indices there occur indices j such that (1) (2) (1) t < j t + q, then, keeping in mind that Cj = Cj + Cj , where Cj = Dj , (2) Cj = Dq+j , we open the brackets and to such a product we put into correspondence the set of terms. For this mapping the images do not intersect. Indeed, let two products F1 and F2 , which dier at least in one factor, be given. If these factors belong to the rst group, then, by virtue of Lemma 3.5, the images are dierent too. If at least one of them belongs to the second group, they dier (1) (2) both in the element Cj and in the element Cj. After opening the brackets, (1) the obtained terms will again dier either in the cofactor Cj or in the cofactor (2) Cj. Using the introduced mapping, we have chosen, in En , some subset X consisting of independent vectors because the set En is such itself (by virtue

of Lemma 3.5) and divided it into nonintersecting subsets Xi. For each xed i let us consider the sums of all elements of the set Xi. It obviously follows that the obtained elements are also linearly independent over R and they coincide with all products of the form Cj1 Cj2 Cjh , where 1 j1 < j2 < < jh m. Dimension of C m is equal to 2m , since the number of all products of this form coincides with the number of all subsets of a set with m elements. Let us now construct (C(f ), g). Assume that C(f ) = C m , g(ei ) = Ci and consider x from E. We have

ai ei ; g(x) =

ai g(ei ) =

ai Ci ;

g (x) =

ai Ci a2 + i

a2 Ci2 = i

a2 Ci2 + i

i=t+q+1

a2 Ci2 i

= f (x, x) = f
a2 In (with (3.4) taken into account); i

i=t+q+1 m m m

ai ei ,

i=1 t i=1 m

ai ei =

i=1 j=1

ai aj f (ei , ej )

a2 (see (1.1)). i

Comparing g (x) and f (x, x) we obtain g 2 (x) = f (x, x)In. Therefore g is correctly dened. Now let us prove that the pair (C m , g) is universal (see Denition 1.2 and its commutative diagram). Let h : E A, where A is an associative R-algebra and h2 (x) = f (x, x)IA. We are to construct a homomorphism of algebras w from C m into A such that h = w g. Let w(Ci ) = h(ei ). Let us continue multiplicatively the homomorphism Ci onto all the products w which are ordered with respect to the growth of indices. Since these products constitute a basis of C m as a R-module, the homomorphism w is uniquely continued over the whole algebra and we have (w g)(ei ) = w(g(ei )) = w(Ci ) = h(ei ), i.e., h = w g. The following statement is valid. Theorem. For the quadratic form (E, f ) satisfying conditions 1) and 2), there exists a matrix Cliord algebra of dimension 2m over R whose generators C1 , C2 ,. , Cm are given by formulas (3.6). Remark 3.1. The generators of the constructed algebra are matrices from Mat R(2m+q ), where m = t + q + k. If q = k = 0, i.e., m = k, then these matrices have the form Ci = Wi Vm , 1 i m, belong to Mat R(2m ) and are generators of a Cliord algebra for the quadratic form (E, f ) satisfying the conditions f (ei , ei ) = 1, 1 i m, f (ei , ej ) = 0, i = j. In such a particular case we can also take, in the role of generators, another system of matrices
+ + + + C1 = W1 Vm1 , C2 = W2 Vm1 ,. , Cm1 = Wm1 Vm1 , Cm = Vm1 ,
belonging to Mat R(2m1 ), i.e., of dimension twice as small. The proof of this statement repeats the above one if the one-to-one correspondence between the sets T1 and T2 is assumed to be as follows: Im Wm1 Vm1 , Wi1 Wih Wm1 Wi1 Wih Vm1 , Wi1 Wih Wi1 Wih Wm1 Vm1 , 1 i1 < < ih < m. Analogously, it can be shown that when t = 1, k = m 1, the order of matrix generators can also be reduced twice. In the role of generators we can take the system of matrices Vm1 , W1 Im1 , W2 Im1 ,. , Wm1 Im1 , belonging to Mat R(2m1 ). Remark 3.2. We would like to note without proving that to construct generators of a Cliord algebra of the quadratic form (E, f ) we can do only with matrices Wi. Namely, as generators Ci we can take the following system of matrices from Mat R(2n ): U3i2 U3i1 U3i , 1 i < t, Ci = U4it U4it3 U4it2 U4it1 , t < i t + q, U t + q < i t + q + k = m, 2t+3q+i , where Ui = Wi In , and n = t + 4q + 3k. Now a few words on innite-dimensional matrix Cliord algebras. Let E be an innite-dimensional free module over R with basis {ei }, i = 1, 2,. , n, and (E, f ) be a form such that f (ei , ej ) = 0 when i = j, and f (ei , ei ) = 1. Consider the algebra of innite-dimensional matrices M , where each matrix in each row and in each column has only a nite number of nonzero elements. Let Wi be the same as above. Consider the matrices Ui = Wi I (I is a unit matrix in M ). They possess the same properties as in Lemmas 3.13.3: Ui2 = I, Ui Uj = Uj Ui (when i = j) and all Bk consist of linearly independent vectors. The algebra that is spanned by them is a Cliord algebra of the form (E, f ). The proof of this fact is analogous to the nite-dimensional case. For the application of innite-dimensional Cliord algebras see [7]. Acknowledgement The paper is partially supported by the 2003 Georgian Academy of Sciences Grant # 1.22.02.

References

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(Received revised 25.06.2004) Authors address: N. Muskhelishvili Institute of Computational Mathematics Georgian Academy of Sciences 8, Akuri St., Tbilisi 0193 Georgia E-mail: udug@gw.acnet.ge