The

generic

dimension

of

a

minimal

realization

of

an

AR-system

Report

J.W.

van

der

Woude

Technische

Universiteit

Delft

Delft

University

of

Technology

Faculteit

der

Technische

Wiskunde

en

Informatica

Faculty

of

Technical

Mathematics

and

Informatics

ISSN

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Delft

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The

generic

dimension

of

a

minimal

realization

of

an

AR-system

J.W.

van

der

Woude

Abstract

In

this

paper

graph

theoretical

methods

are

developed

for

the

determination

of

the

generic

dimension

of

a

minimal

realization

of

a

so-called

AR-system

only

using

the

structure

present

in

the

AR-equations

Introduction

In

this

paper

we

follow

the

approach

towards

linear

systems

initiated

by

J.C.

Willems

-LRB-

cf.

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

One

of

the

features

of

this

approach

is

that

no

a

priori

distinction

is

made

between

inputs

and

outputs

but

that

they

both

are

part

of

the

so-called

external

variable

In

this

paper

we

start

from

systems

described

by

means

of

equations

in

the

external

variable

w

-LRB-

t

-RRB-

The

equations

that

we

consider

are

linear

homogeneous

difference

equations

with

constant

coefficients

The

idea

is

that

often

when

modelling

a

dynamical

phenomenon

we

may

start

by

writing

down

equations

and

by

collecting

all

variables

into

a

single

vector

The

equations

may

for

instance

come

from

physical

laws

and

given

relationships

The

model

obtained

is

called

an

AR-model

The

coefficients

involved

in

the

equations

often

have

values

that

are

not

known

exactly

but

it

is

known

where

they

occur

within

the

equations

Having

not

yet

made

a

distinction

between

inputs

and

outputs

the

next

step

in

the

modelling

process

can

be

the

splitting

of

the

external

variable

into

an

input

and

an

output

in

such

a

way

that

the

transfer

matrix

from

the

input

to

the

output

exists

and

is

proper

Such

a

splitting

is

always

possible

and

will

be

called

in

this

paper

a

causal

splitting

Next

it

may

be

desired

to

actually

compute

the

previous

mentioned

transfer

matrix

and

to

realize

it

by

means

of

a

realization

with

a

state

space

of

minimal

dimension

Such

a

minimal

realization

of

the

transfer

matrix

corresponding

to

a

causal

splitting

of

the

external

variable

will

be

referred

to

in

this

paper

just

as

a

minimal

realization

and

its

dimension

simply

will

be

the

dimension

of

its

state

space

Clearly

the

dimension

of

a

minimal

realization

is

an

integer

of

great

interest

In

general

this

dimension

will

depend

on

the

value

and

the

location

of

the

coefficients

in

the

equations

The

purpose

of

this

paper

is

to

show

however

that

generically

this

dimension

depends

only

on

the

largest

and

smallest

time-shifts

of

every

variable

in

each

of

the

equations

Hence

generically

only

the

location

of

the

coefficients

within

in

the

equations

is

important

for

the

generic

dimension

of

a

minimal

realization

and

not

their

numerical

value

Here

the

word

generic

indicates

that

the

dimension

of

a

minimal

realization

will

be

the

same

for

almost

all

possible

coefficient

values

In

this

paper

we

present

an

algorithm

to

determine

this

generic

dimension

using

so-called

bipartite

graphs

and

matchings

In

this

paper

we

occasionally

think

of

the

coefficients

in

the

equations

as

independent

free

parameters

In

-LSB-

-RSB-

it

has

been

indicted

how

a

causal

splitting

of

the

external

variable

can

be

obtained

in

case

only

the

location

of

the

free

parameters

is

known

The

approach

in

-LSB-

-RSB-

is

based

on

-LSB-

-RSB-

and

is

more

general

than

the

approach

in

the

present

paper

This

is

caused

by

the

fact

that

in

the

approach

in

-LSB-

-RSB-

and

-LSB-

-RSB-

in

addition

to

the

free

parameters

also

fixed

constants

are

allowed

in

the

equations

Fixed

constants

are

coefficients

that

have

a

known

fixed

value

For

instance

the

coefficients

and

that

often

occur

after

the

introduction

of

new

variables

and

the

rewriting

of

equations

can

be

seen

as

fixed

constants

A

drawback

of

the

more

general

approach

in

-LSB-

-RSB-

and

-LSB-

-RSB-

is

that

the

obtained

results

can

not

be

presented

in

a

nice

intuitive

way

with

bipartite

graphs

and

matchings

as

is

the

case

when

only

free

parameters

are

allowed

The

same

situation

arises

here

Also

now

we

can

look

for

equations

that

both

contain

free

parameters

and

fixed

constants

The

results

however

get

technically

involved

and

are

not

so

attractive

as

when

dealing

with

only

free

parameters

in

the

equations

For

this

reason

we

concentrate

here

on

the

case

that

the

equations

only

contain

coefficients

occasionally

considered

as

independent

free

parameters

Polynomials

As

in

-LSB-

-RSB-

part

I

we

shall

be

working

with

polynomials

in

s

and

with

polynomials

in

s

and

s

Below

we

review

some

aspects

of

these

polynomials

and

we

use

the

following

notation

If

Xg

are

matrices

all

with

the

same

number

of

columns

we

denote

col

-LRB-

Xg

-RRB-

for

the

matrix

obtained

by

stacking

Xg

on

top

of

each

other

i.e.

col

-LRB-

Xg

-RRB-

BBBB

Xg

CCCCA

We

start

with

the

usual

type

of

polynomials

R

-LSB-

s

-RSB-

We

denote

the

set

of

polynomials

with

real

coefficients

in

the

indeterminate

s

by

R

-LSB-

s

-RSB-

We

write

Rg

-LSB-

s

-RSB-

for

the

set

of

g-vectors

with

entries

in

R

-LSB-

s

-RSB-

and

Rg?q

-LSB-

s

-RSB-

for

the

set

of

g?q

matrices

with

entries

in

R

-LSB-

s

-RSB-

Elements

of

Rg

-LSB-

s

-RSB-

are

called

polynomial

vectors

and

elements

of

Rg?q

-LSB-

s

-RSB-

are

called

polynomial

matrices

For

a

scalar

polynomial

p

-LRB-

s

-RRB-

R

-LSB-

s

-RSB-

or

a

polynomial

row

p

-LRB-

s

-RRB-

-LSB-

s

-RSB-

p

-LRB-

s

-RRB-

we

define

the

-LRB-

leading

-RRB-

degree

of

p

-LRB-

s

-RRB-

to

be

the

exponent

of

the

highest

power

in

s

present

in

p

-LRB-

s

-RRB-

We

denote

this

degree

by

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

It

is

clear

that

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

If

p

-LRB-

s

-RRB-

we

define

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

to

make

sure

that

the

degree

of

the

product

of

any

two

scalar

polynomials

equals

the

sum

of

their

degrees

A

polynomial

matrix

U

-LRB-

s

-RRB-

Rg?g

-LSB-

s

-RSB-

is

called

a

unimodular

matrix

if

it

has

an

inverse

in

the

set

Rg?g

-LSB-

s

-RSB-

It

is

easy

to

see

that

U

-LRB-

s

-RRB-

Rg?g

-LSB-

s

-RSB-

is

unimodular

if

and

only

if

det

U

-LRB-

s

-RRB-

with

a

nonzero

constant

-LRB-

det

stands

for

determinant

-RRB-

This

follows

easily

from

the

Cramer

formula

and

the

fact

that

the

only

units

-LRB-

invertible

elements

with

respect

to

multiplication

-RRB-

in

R

-LSB-

s

-RSB-

are

nonzero

constants

Let

P

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

pg

-LRB-

s

-RRB-

-RRB-

be

a

polynomial

matrix

in

Rg?q

-LSB-

s

-RSB-

with

rows

-LRB-

s

-RRB-

-LRB-

s

-RRB-

pg

-LRB-

s

-RRB-

-LSB-

s

-RSB-

Write

pi

-LRB-

s

-RRB-

pini

sni

pini

with

for

i

g

deg

-LRB-

pi

-LRB-

s

-RRB-

-RRB-

ni

and

pij

for

j

ni

Then

the

constant

g

q-matrix

L

col

-LRB-

pgng

-RRB-

denotes

the

leading

row

coefficient

matrix

of

P

-LRB-

s

-RRB-

The

polynomial

matrix

P

-LRB-

s

-RRB-

is

called

row

reduced

if

its

leading

row

coefficient

matrix

has

full

row

rank

It

has

been

shown

in

-LSB-

-RSB-

-LRB-

see

also

-LSB-

-RSB-

part

I

-RRB-

that

if

P

-LRB-

s

-RRB-

is

a

full

row

rank

matrix

there

exists

a

unimodular

matrix

U

-LRB-

s

-RRB-

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

is

row

reduced

Further

it

is

well-known

that

if

P

-LRB-

s

-RRB-

Rg?q

-LSB-

s

-RSB-

has

rank

p

there

is

a

unimodular

matrix

U

-LRB-

s

-RRB-

Rg?g

-LSB-

s

-RSB-

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

col

-LRB-

P

-LRB-

s

-RRB-

-RRB-

with

P

-LRB-

s

-RRB-

Rp?q

-LSB-

s

-RSB-

a

full

row

rank

matrix

From

this

it

follows

that

if

P

-LRB-

s

-RRB-

Rg?q

-LSB-

s

-RSB-

and

rank

P

-LRB-

s

-RRB-

p

there

is

a

unimodular

matrix

U

-LRB-

s

-RRB-

Rg?g

-LSB-

s

-RSB-

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

col

-LRB-

P

-LRB-

s

-RRB-

-RRB-

with

P

-LRB-

s

-RRB-

Rp?q

-LSB-

s

-RSB-

and

P

-LRB-

s

-RRB-

is

row

reduced

Finally

let

P

-LRB-

s

-RRB-

be

a

full

row

rank

polynomial

in

Rp?q

-LSB-

s

-RSB-

and

U

-LRB-

s

-RRB-

Rp?p

-LSB-

s

-RSB-

a

unimodular

matrix

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

is

row

reduced

with

row

degrees

np

Then

it

follows

easily

with

the

Binet-Cauchy

formula

-LRB-

cf.

-LSB-

-RSB-

-RRB-

-LRB-

see

also

later

-RRB-

that

pX

i

ni

maxfdeg

-LRB-

f

-LRB-

s

-RRB-

-RRB-

jf

-LRB-

s

-RRB-

is

a

p-th

order

minor

of

P

-LRB-

s

-RRB-

g

-LRB-

a

p-th

order

minor

of

P

-LRB-

s

-RRB-

is

the

determinant

of

a

p

p

submatrix

of

P

-LRB-

s

-RRB-

-RRB-

R

-LSB-

s

s

-RSB-

We

denote

R

-LSB-

s

s

-RSB-

for

the

set

of

polynomials

-LRB-

finite

Laurent

series

-RRB-

with

real

coefficients

in

the

indeterminates

s

and

s

We

write

Rg

-LSB-

s

s

-RSB-

for

the

set

of

g-vectors

with

entries

in

R

-LSB-

s

s

-RSB-

and

Rg?q

-LSB-

s

s

-RSB-

for

the

set

of

g

q-matrices

with

entries

in

R

-LSB-

s

s

-RSB-

We

also

call

the

elements

of

Rg

-LSB-

s

s

-RSB-

polynomial

vectors

and

the

elements

of

Rg?q

-LSB-

s

s

-RSB-

polynomial

matrices

For

a

scalar

polynomial

p

-LRB-

s

-RRB-

R

-LSB-

s

s

-RSB-

or

a

polynomial

row

p

-LRB-

s

-RRB-

-LSB-

s

s

-RSB-

p

-LRB-

s

-RRB-

we

define

the

leading

degree

of

p

-LRB-

s

-RRB-

to

be

the

exponent

of

the

highest

power

in

s

present

in

p

-LRB-

s

-RRB-

and

we

denote

this

degree

by

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

In

the

same

spirit

we

define

the

trailing

degree

of

p

-LRB-

s

-RRB-

to

be

the

exponent

of

the

lowest

power

in

s

present

in

p

-LRB-

s

-RRB-

This

degree

we

denote

by

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

Note

that

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

and

that

both

degrees

can

be

negative

If

p

-LRB-

s

-RRB-

we

define

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

and

deg

-LRB-

p

-LRB-

s

-RRB-

-RRB-

again

making

sure

that

the

leading

-LRB-

trailing

-RRB-

degree

of

the

product

of

any

two

scalar

polynomials

equals

the

sum

of

their

leading

-LRB-

trailing

-RRB-

degrees

A

polynomial

matrix

U

-LRB-

s

-RRB-

Rg?g

-LSB-

s

s

-RSB-

is

called

a

bilaterally

unimodular

matrix

if

it

has

an

inverse

in

the

set

Rg?g

-LSB-

s

s

-RSB-

The

units

in

the

set

R

-LSB-

s

s

-RSB-

are

elements

of

the

form

ffst

with

a

nonzero

constant

and

t

an

integer

This

explains

that

U

-LRB-

s

-RRB-

Rg?g

-LSB-

s

s

-RSB-

is

bilaterally

unimodular

if

and

only

if

det

U

-LRB-

s

-RRB-

ffst

with

and

t

integer

Note

that

there

exist

bilaterally

unimodular

matrices

that

just

have

a

constant

nonzero

determinant

Indeed

let

U

-LRB-

s

-RRB-

be

a

bilaterally

unimodular

matrix

with

det

U

-LRB-

s

-RRB-

ffst

and

t

integer

Select

a

row

-LRB-

or

cloumn

-RRB-

in

U

-LRB-

s

-RRB-

and

multiply

it

with

s?t

to

obtain

the

matrix

U

-LRB-

s

-RRB-

Then

U

-LRB-

s

-RRB-

is

bilaterally

unimodular

matrix

and

det

U

-LRB-

s

-RRB-

Now

let

P

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

pg

-LRB-

s

-RRB-

-RRB-

Rg?q

-LSB-

s

s

-RSB-

and

write

pi

-LRB-

s

-RRB-

pini

sni

pini

pimi

smi

pimi

smi

with

for

i

g

deg

-LRB-

pi

-LRB-

s

-RRB-

-RRB-

ni

mi

deg

-LRB-

pi

-LRB-

s

-RRB-

-RRB-

and

pij

for

j

mi

ni

We

denote

the

leading

row

coefficient

matrix

of

P

-LRB-

s

-RRB-

by

L

col

-LRB-

pgng

-RRB-

and

the

trailing

row

coefficient

matrix

by

L

col

-LRB-

pgmg

-RRB-

The

matrix

P

-LRB-

s

-RRB-

Rg?q

-LSB-

s

s

-RSB-

is

called

bilaterally

row

reduced

if

both

its

leading

row

coefficient

matrix

and

its

trailing

row

coefficient

matrix

have

full

row

rank

It

has

been

indicated

in

-LSB-

-RSB-

part

I

that

if

P

-LRB-

s

-RRB-

is

a

full

row

rank

matrix

there

exists

a

bilaterally

unimodular

matrix

U

-LRB-

s

-RRB-

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

is

bilaterally

row

reduced

In

the

same

way

as

above

it

can

be

shown

that

for

P

-LRB-

s

-RRB-

Rg?q

-LSB-

s

s

-RSB-

with

rank

P

-LRB-

s

-RRB-

p

there

is

a

bilaterally

unimodular

matrix

U

-LRB-

s

-RRB-

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

col

-LRB-

P

-LRB-

s

-RRB-

-RRB-

with

P

-LRB-

s

-RRB-

Rp?q

-LSB-

s

s

-RSB-

bilaterally

row

reduced

In

the

latter

the

bilaterally

unimodular

matrix

U

-LRB-

s

-RRB-

may

even

be

such

that

det

U

-LRB-

s

-RRB-

As

far

as

the

last

remark

of

the

previous

subsection

is

concerned

a

similar

remark

holds

for

polynomial

matrices

in

s

and

s

Let

P

-LRB-

s

-RRB-

be

a

full

row

rank

polynomial

in

Rp?q

-LSB-

s

s

-RSB-

and

let

U

-LRB-

s

-RRB-

Rp?p

-LSB-

s

s

-RSB-

a

be

bilaterally

unimodular

matrix

with

det

U

-LRB-

s

-RRB-

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

is

bilaterally

row

reduced

with

leading

row

degrees

np

and

trailing

row

degrees

mp

Then

it

follows

with

the

Binet-Cauchy

formula

that

pX

i

ni

maxfdeg

-LRB-

f

-LRB-

s

-RRB-

-RRB-

jf

-LRB-

s

-RRB-

is

a

p-th

order

minor

of

P

-LRB-

s

-RRB-

g

and

pX

i

mi

minfdeg

-LRB-

f

-LRB-

s

-RRB-

-RRB-

jf

-LRB-

s

-RRB-

is

a

p-th

order

minor

of

P

-LRB-

s

-RRB-

g

Moreover

if

U

-LRB-

s

-RRB-

Rp?p

-LSB-

s

s

-RSB-

is

just

a

bilaterally

unimodular

matrix

-LRB-

with

det

U

-LRB-

s

-RRB-

ffst

for

some

and

integer

t

-RRB-

such

that

U

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

is

bilaterally

row

reduced

with

leading

row

degrees

and

trailing

row

degrees

Then

it

follows

easily

that

Ppi

ni

Ppi

mi

Ppi

-LRB-

ni

mi

-RRB-

Ppi

-LRB-

-RRB-

Ppi

Ppi

The

dimension

of

a

minimal

realization

As

announced

above

we

study

systems

described

by

means

of

a

finite

number

of

linear

difference

equations

with

constant

coefficients

The

equations

are

in

the

external

variable

w

-LRB-

t

-RRB-

Rq

We

write

ri

-LRB-

oe

-RRB-

w

-LRB-

t

-RRB-

for

the

i-th

equation

Here

ri

-LRB-

s

-RRB-

is

a

q

polynomial

matrix

in

the

indeterminate

s

and

oe

denotes

the

shift

operator

i.e.

-LRB-

oef

-RRB-

-LRB-

t

-RRB-

f

-LRB-

t

-RRB-

The

total

system

is

described

by

R

-LRB-

oe

-RRB-

w

-LRB-

t

-RRB-

with

R

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

rg

-LRB-

s

-RRB-

-RRB-

Rg?q

-LSB-

s

-RSB-

Following

-LSB-

-RSB-

part

I

we

call

R

-LRB-

oe

-RRB-

w

-LRB-

t

-RRB-

an

AR-description

and

we

make

a

distinction

between

the

case

that

the

time

axis

is

T

Z

-LRB-

the

nonnegative

integers

-RRB-

and

the

case

that

T

Z

-LRB-

the

integers

-RRB-

As

indicated

before

an

AR-description

can

be

seen

as

a

first

step

in

the

process

of

modelling

a

dynamical

phenomenon

The

next

step

may

be

the

computation

of

a

causal

splitting

i.e.

a

splitting

of

the

external

variable

w

-LRB-

t

-RRB-

into

an

input

u

-LRB-

t

-RRB-

and

an

output

y

-LRB-

t

-RRB-

such

that

the

transfer

matrix

from

the

input

to

the

output

is

a

proper

rational

matrix

This

step

can

be

followed

by

the

actual

computation

of

this

transfer

matrix

and

by

realizing

it

by

means

of

a

minimal

realization

Details

of

these

steps

can

be

found

in

-LSB-

-RSB-

part

I

In

this

section

we

give

a

method

to

directly

determine

the

dimension

of

the

minimal

realization

from

the

underlying

AR-description

without

explicitely

performing

the

above

steps

The

starting

point

is

the

AR-description

R

-LRB-

oe

-RRB-

w

-LRB-

t

-RRB-

with

rank

R

-LRB-

s

-RRB-

=p

We

distinguish

between

the

case

that

T

Z

and

the

case

that

T

Z

Following

the

proofs

in

-LSB-

-RSB-

part

I

the

next

algorithms

are

immediate

Algorithm

to

compute

the

dimension

of

a

minimal

realization

T

Z

Take

a

unimodular

matrix

U

-LRB-

s

-RRB-

g

-LSB-

s

-RSB-

such

that

U

-LRB-

s

-RRB-

R

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-RRB-

with

-LRB-

s

-RRB-

a

p

q

row

reduced

polynomial

matrix

Let

the

degrees

of

the

rows

of

-LRB-

s

-RRB-

be

np

Then

the

dimension

n

of

a

minimal

realization

is

given

by

Ppi

ni

It

follows

immediately

from

the

previous

that

n

maxfdeg

-LRB-

f

-LRB-

s

-RRB-

-RRB-

jf

-LRB-

s

-RRB-

is

a

p-th

order

minor

of

-LRB-

s

-RRB-

g

T

Z

Take

bilaterally

unimodular

matrix

U

-LRB-

s

-RRB-

Rg?g

-LSB-

s

s

-RSB-

such

that

U

-LRB-

s

-RRB-

R

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-RRB-

with

-LRB-

s

-RRB-

a

p

q

bilaterally

row

reduced

polynomial

matrix

Let

the

leading

degrees

of

the

rows

of

-LRB-

s

-RRB-

be

np

and

the

trailing

degrees

mp

Then

the

dimension

n

of

a

minimal

realization

is

given

by

Ppi

ni

Ppi

mi

Ppi

-LRB-

ni

mi

-RRB-

Also

here

it

is

immediate

that

if

det

U

-LRB-

s

-RRB-

then

n

n

n

with

n

maxfdeg

-LRB-

f

-LRB-

s

-RRB-

-RRB-

jf

-LRB-

s

-RRB-

is

a

p-th

order

minor

of

-LRB-

s

-RRB-

g

and

n

minfdeg

-LRB-

f

-LRB-

s

-RRB-

-RRB-

jf

-LRB-

s

-RRB-

is

a

p-th

order

minor

of

-LRB-

s

-RRB-

g

Direct

method

In

the

previous

section

we

described

a

direct

method

for

the

computation

of

the

dimension

of

a

minimal

realization

of

the

transfer

matrix

corresponding

a

causal

splitting

of

the

external

variable

The

computation

involved

the

reduction

of

the

matrix

R

-LRB-

s

-RRB-

to

-LRB-

bilaterally

-RRB-

row

reduced

form

In

the

present

section

we

present

an

alternative

method

to

compute

this

minimal

state

dimension

The

method

is

expressed

entirely

in

terms

of

-LRB-

submatrices

of

-RRB-

the

matrix

R

-LRB-

s

-RRB-

and

does

not

require

the

reduction

to

some

special

form

The

reason

for

introducing

this

alternative

method

is

that

we

then

can

incorporate

the

assumption

that

the

value

of

the

nonzero

coeficients

in

the

matrix

R

-LRB-

s

-RRB-

are

not

known

but

only

their

locations

The

starting

point

is

the

AR-description

R

-LRB-

oe

-RRB-

w

-LRB-

t

-RRB-

with

rank

R

-LRB-

s

-RRB-

=p

We

mainly

treat

the

case

that

T

Z

The

case

that

T

Z

will

be

discussed

shortly

Let

U

-LRB-

s

-RRB-

g

-LSB-

s

-RSB-

be

a

unimodular

matrix

such

that

U

-LRB-

s

-RRB-

R

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-RRB-

with

-LRB-

s

-RRB-

a

p

q

row

reduced

polynomial

matrix

Denote

the

row

degrees

of

the

-LRB-

s

-RRB-

by

np

and

write

n

np

Write

Rp?q

for

the

leading

coefficient

matrix

of

-LRB-

s

-RRB-

Then

rank

p

and

therefore

contains

a

nonsingular

p?p

submatrix

Assume

that

is

composed

of

the

columns

of

with

index

ip

Denote

P

-LRB-

s

-RRB-

for

the

polynomial

matrix

made

up

of

the

columns

of

-LRB-

s

-RRB-

with

index

ip

and

P

-LRB-

s

-RRB-

for

the

polynomial

matrix

made

up

of

the

columns

of

R

-LRB-

s

-RRB-

with

index

ip

So

P

-LRB-

s

-RRB-

Rp?p

-LRB-

s

-RRB-

and

P

-LRB-

s

-RRB-

Rg?p

-LRB-

s

-RRB-

Note

that

det

P

-LRB-

s

-RRB-

np

det

lower

order

terms

in

s.

Hence

deg

-LRB-

det

P

-LRB-

s

-RRB-

-RRB-

n.

Let

be

any

p

p

submatrix

of

consisting

of

the

columns

with

index

jp

Let

M

-LRB-

s

-RRB-

and

M

-LRB-

s

-RRB-

be

the

p

p

submatrices

of

-LRB-

s

-RRB-

and

R

-LRB-

s

-RRB-

respectively

consisting

of

the

columns

with

index

jp

As

before

it

follows

that

det

M

-LRB-

s

-RRB-

sn

det

lower

order

terms

in

s.

Hence

deg

-LRB-

det

M

-LRB-

s

-RRB-

-RRB-

n

Now

we

return

to

P

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

and

and

we

claim

that

P

-LRB-

s

-RRB-

is

a

greatest

right

divisor

of

P

-LRB-

s

-RRB-

We

mean

by

this

that

P

-LRB-

s

-RRB-

is

a

right

divisor

of

P

-LRB-

s

-RRB-

i.e.

P

-LRB-

s

-RRB-

V

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

for

some

polynomial

matrix

V

-LRB-

s

-RRB-

and

any

other

right

divisor

of

P

-LRB-

s

-RRB-

is

also

a

right

divisor

of

P

-LRB-

s

-RRB-

To

prove

the

claim

we

denote

U

-LRB-

s

-RRB-

-LSB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-RSB-

with

-LRB-

s

-RRB-

Rg?p

-LRB-

s

-RRB-

and

-LRB-

s

-RRB-

Rg

-LRB-

q?p

-RRB-

-LRB-

s

-RRB-

U

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-RRB-

with

-LRB-

s

-RRB-

Rp?g

-LRB-

s

-RRB-

and

-LRB-

s

-RRB-

R

-LRB-

g?p

-RRB-

g

-LRB-

s

-RRB-

and

we

let

P

-LRB-

s

-RRB-

Rt?p

-LSB-

s

-RSB-

be

some

right

divisor

of

P

-LRB-

s

-RRB-

i.e.

there

is

a

polynomial

matrix

V

-LRB-

s

-RRB-

Rg?t

-LSB-

s

-RSB-

such

that

P

-LRB-

s

-RRB-

V

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

First

note

that

P

-LRB-

s

-RRB-

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

implying

that

P

-LRB-

s

-RRB-

is

a

right

divisor

of

P

-LRB-

s

-RRB-

Further

it

follows

that

P

-LRB-

s

-RRB-

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

-LRB-

s

-RRB-

V

-LRB-

s

-RRB-

P

-LRB-

s

-RRB-

The

latter

implies

that

any

right

divisor

of

P

-LRB-

s

-RRB-

also

is

a

right

divisor

of

P

-LRB-

s

-RRB-

Therefore

P

-LRB-

s

-RRB-

is

a

greatest

right

divisor

of

P

-LRB-

s

-RRB-

This

proves

our

claim

It

is

clear

that

any

two

greatest

right

divisors

of

P

-LRB-

s

-RRB-

can

be

obtained

from

each

other

by

left

multiplication

with

a

unimodular

matrix

We

refer

to

any

square

greatest

right

divisor

of

P

-LRB-

s

-RRB-

as

grd

-LRB-

P

-LRB-

s

-RRB-

-RRB-

With

this

notation

it

is

immediate

that

deg

-LRB-

det

-LRB-

P

-LRB-

s

-RRB-

-RRB-

-RRB-

deg

-LRB-

det

-LRB-

grd

-LRB-

P

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

n.

Returning

to

M

-LRB-

s

-RRB-

M

-LRB-

s

-RRB-

and

it

follows

similarly

that

M

-LRB-

s

-RRB-

is

a

square

greatest

right

divisor

of

M

-LRB-

s

-RRB-

implying

that

deg

-LRB-

det

-LRB-

M

-LRB-

s

-RRB-

-RRB-

-RRB-

deg

-LRB-

det

-LRB-

grd

-LRB-

M

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

n.

Summarizing

the

previous

we

obtain

the

following

To

determine

the

dimension

n

of

a

minimal

realization

starting

from

the

AR-matrix

R

-LRB-

s

-RRB-

with

rank

p

we

can

compute

the

maximum

degree

of

the

determinant

of

a

square

greatest

right

divisor

of

a

matrix

composed

of

p

columns

of

R

-LRB-

s

-RRB-

Below

we

will

state

this

in

more

matematical

terms

Therefore

we

denote

V

gg

for

the

set

of

row

indices

and

W

qg

for

the

set

of

column

indices

of

R

-LRB-

s

-RRB-

Further

for

any

set

S

we

denote

jSj

for

the

number

of

elements

it

consists

of

If

I

V

and

J

W

we

denote

RIJ

-LRB-

s

-RRB-

for

the

matrix

consisting

of

the

elements

of

R

-LRB-

s

-RRB-

with

row

index

in

I

and

column

index

in

J

Then

the

above

implies

that

n

max

J?Wp

deg

-LRB-

det

-LRB-

grd

-LRB-

RVJ

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

where

Wp

fJ

W

jJ

j

pg

is

the

collection

of

index

sets

in

W

consisting

of

p

distinct

elements

Hence

in

case

T

Z

we

can

compute

the

dimension

of

a

minimal

realization

by

the

above

formula

A

similar

formula

can

be

derived

in

case

T

Z

In

that

case

the

dimension

n

can

be

written

as

n

n

where

n

is

equal

to

n

and

n

is

computed

as

above

with

deg

replaced

by

deg

and

maximization

replaced

by

minimization

To

see

why

the

latter

is

true

consider

the

matrix

R

-LRB-

s

-RRB-

to

be

an

element

of

Rg?g

-LSB-

s

s

-RSB-

Further

let

U

-LRB-

s

-RRB-

g

-LSB-

s

s

-RSB-

with

det

U

-LRB-

s

-RRB-

be

a

bilaterally

unimodular

matrix

such

that

U

-LRB-

s

-RRB-

R

-LRB-

s

-RRB-

col

-LRB-

-LRB-

s

-RRB-

-RRB-

with

-LRB-

s

-RRB-

a

p

q-bilaterally

row

reduced

polynomial

matrix

Then

following

the

above

reasoning

it

is

easily

seen

that

n

n

where

the

notion

of

greatest

right

divisor

has

to

be

adapted

to

be

valid

for

matrices

with

elements

in

R

-LSB-

s

s

-RSB-

Next

observe

that

the

leading

and

trailing

degrees

are

opposite

notions

in

the

sense

that

deg

p

-LRB-

s

-RRB-

deg

p

-LRB-

s

-RRB-

for

any

element

p

-LRB-

s

-RRB-

R

-LSB-

s

s

-RSB-

This

means

that

the

sum

of

the

trailing

row

degrees

of

a

full

row

rank

matrix

R

-LRB-

s

-RRB-

equals

minus

the

sum

of

the

leading

row

degrees

of

the

matrix

R

-LRB-

s

-RRB-

The

latter

can

be

computed

by

means

of

considering

deg

and

by

maximizing

-LRB-

see

before

-RRB-

implying

that

the

former

can

be

computed

by

considering

deg

and

by

minimizing

The

obtained

results

are

summarized

in

the

next

theorem

Theorem

a

-RRB-

For

T

Z

a

minimal

realization

has

dimension

n

where

n

max

J?Wp

deg

-LRB-

det

-LRB-

grd

-LRB-

RVJ

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

b

-RRB-

For

T

Z

a

minimal

realization

has

dimension

n

n

n

where

n

max

J?Wp

deg

-LRB-

det

-LRB-

grd

-LRB-

RVJ

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

n

min

J?Wp

deg

-LRB-

det

-LRB-

grd

-LRB-

RVJ

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

We

conclude

this

section

with

some

remarks

that

will

be

used

in

the

next

section

Therefore

consider

a

polynomial

matrix

R

-LRB-

s

-RRB-

that

has

full

column

rank

and

is

written

as

R

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

with

-LRB-

s

-RRB-

square

and

invertible

It

follows

that

-LRB-

s

-RRB-

has

full

column

rank

and

it

is

easily

seen

that

a

greatest

right

divisor

of

R

-LRB-

s

-RRB-

can

be

given

by

-LRB-

s

-RRB-

-LRB-

s

-RRB-

grd

-LRB-

-LRB-

s

-RRB-

-RRB-

where

grd

-LRB-

-LRB-

s

-RRB-

-RRB-

denotes

a

square

greatest

right

divisor

of

-LRB-

s

-RRB-

Further

it

follows

that

det

-LRB-

grd

-LRB-

R

-LRB-

s

-RRB-

-RRB-

-RRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

det

-LRB-

grd

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

Bipartite

graphs

and

matchings

In

this

section

we

consider

polynomial

matrices

of

which

the

coefficients

are

unknown

and

may

have

any

real

value

We

therefore

can

think

of

these

coefficients

as

independent

free

parameters

We

relate

these

polynomial

matrices

with

bipartite

graphs

and

show

how

the

generic

value

of

the

rank

and

some

determinantal

degrees

can

be

computed

by

graph

theroretic

methods

We

consider

the

polynomial

matrix

R

-LRB-

s

-RRB-

Rg?q

-LSB-

s

-RSB-

and

we

assume

that

we

only

know

its

structure

With

this

we

mean

that

of

each

entry

of

R

-LRB-

s

-RRB-

we

know

which

power

of

s

are

present

but

we

do

not

know

the

coefficients

by

which

the

powers

are

multiplied

In

fact

it

turns

out

that

we

only

have

to

know

the

exponents

of

the

highest

and

lowest

powers

of

s

of

each

nonzero

entry

of

R

-LRB-

s

-RRB-

For

instance

an

entry

of

R

-LRB-

s

-RRB-

may

be

of

the

form

s

The

values

of

and

may

be

unknown

and

it

is

not

required

to

know

whether

the

entry

contains

terms

with

or

Further

R

-LRB-

s

-RRB-

may

contain

entries

that

are

identically

equal

to

zero

No

powers

of

s

are

present

in

these

entries

and

consequently

no

coefficients

Given

the

matrix

R

-LRB-

s

-RRB-

described

above

we

can

construct

a

bipartite

graph

denoted

B

The

graph

B

consists

of

the

sets

V

and

W

of

vertices

and

the

set

A

of

arcs

directed

from

W

to

V

We

therefore

also

denote

B

-LRB-

V

W

A

-RRB-

More

in

detail

for

R

-LRB-

s

-RRB-

Rg?q

-LSB-

s

-RSB-

we

have

V

gg

W

qg

and

A

f

-LRB-

j

i

-RRB-

jRij

-LRB-

s

-RRB-

g

Here

-LRB-

j

i

-RRB-

denotes

the

arc

from

vertex

j

W

to

vertex

i

V

and

Rij

-LRB-

s

-RRB-

indicates

that

the

-LRB-

i

j

-RRB-

entry

of

R

-LRB-

s

-RRB-

is

not

identically

equal

to

zero

Note

that

in

the

previous

sections

V

was

referred

to

as

the

set

of

row

indices

and

W

as

the

set

of

column

indices

of

the

matrix

R

-LRB-

s

-RRB-

An

example

of

a

bipartite

graph

is

given

in

the

next

section

Below

we

shall

need

weights

on

the

arcs

of

B

These

weights

are

defined

by

two

weight

functions

ffi

ffi

A

Z

given

as

follows

For

-LRB-

j

i

-RRB-

A

we

define

ffi

-LRB-

j

i

-RRB-

deg

-LRB-

Rij

-LRB-

s

-RRB-

-RRB-

and

ffi

-LRB-

j

i

-RRB-

deg

-LRB-

Rij

-LRB-

s

-RRB-

-RRB-

As

before

deg

-LRB-

Rij

-LRB-

s

-RRB-

-RRB-

denotes

the

exponent

of

the

highest

power

of

s

in

Rij

-LRB-

s

-RRB-

and

deg

-LRB-

Rij

-LRB-

s

-RRB-

-RRB-

denotes

the

exponent

of

the

lowest

power

of

s

in

Rij

-LRB-

s

-RRB-

If

an

arc

a

A

is

directed

from

vertex

j

W

to

the

vertex

i

V

we

also

denote

a

-LRB-

j

i

-RRB-

The

vertex

j

W

is

then

called

the

initial

vertex

and

the

vertex

i

V

is

called

the

terminal

vertex

of

a

A

matching

in

the

graph

B

-LRB-

V

W

A

-RRB-

is

a

subset

N

of

A

consisting

of

arcs

that

do

not

have

any

vertices

in

common

Hence

the

number

of

arcs

in

N

equals

the

number

of

initial

vertices

of

the

arcs

in

N

and

also

the

number

of

terminal

vertices

of

the

arcs

in

N

The

number

of

arcs

in

a

matching

N

is

called

the

order

of

the

matching

It

is

clear

that

in

the

graph

B

there

always

exists

a

matching

with

maximum

order

Given

the

arc

weight

functions

ffi

and

ffi

we

define

the

ffi

weight

of

a

matching

to

be

the

sum

of

the

ffi

weights

of

the

arcs

the

matching

is

composed

of

and

similarly

for

the

ffi

weight

of

a

matching

Given

the

collection

of

maximum

order

matchings

there

are

maximum

order

matchings

that

have

maximum

ffi

weight

and

maximum

order

matchings

that

have

minimum

ffi

weight

For

numerically

specified

polynomial

matrices

H

-LRB-

s

-RRB-

we

say

that

rank

H

-LRB-

s

-RRB-

r

if

there

exists

a

minor

of

H

-LRB-

s

-RRB-

of

order

r

that

is

nonzero

while

every

minor

of

H

-LRB-

s

-RRB-

of

order

r

is

identically

equal

to

zero

As

we

have

here

a

polynomial

matrix

R

-LRB-

s

-RRB-

of

which

the

coefficients

are

unknown

we

need

a

different

notion

of

rank

that

in

principle

not

depends

on

the

values

of

the

coefficients

We

therefore

say

that

generically

rank

R

-LRB-

s

-RRB-

p

-LRB-

or

the

generic

rank

R

-LRB-

s

-RRB-

equals

p

-RRB-

if

the

identity

rank

R

-LRB-

s

-RRB-

p

holds

for

almost

all

possible

values

of

the

coefficients

present

in

R

-LRB-

s

-RRB-

Here

almost

all

is

to

be

understood

as

for

all

except

for

those

in

some

set

with

zero

Lebesgue

measure

We

are

now

in

position

to

state

the

next

results

Theorem

The

generic

rank

of

R

-LRB-

s

-RRB-

equals

the

maximum

among

all

the

orders

of

matchings

in

B.

Assume

that

g

q

p

with

p

the

generic

rank

of

R

-LRB-

s

-RRB-

The

assumption

g

q

p

means

that

R

-LRB-

s

-RRB-

is

square

and

generically

nonsingular

Theorem

Let

g

q

p

Then

deg

-LRB-

det

-LRB-

R

-LRB-

s

-RRB-

-RRB-

-RRB-

equals

the

maximum

among

the

ffi

weights

of

all

maximum

order

matchings

in

B

and

deg

-LRB-

det

-LRB-

R

-LRB-

s

-RRB-

-RRB-

-RRB-

equals

the

minimum

among

the

ffi

weights

of

all

maximum

order

matchings

in

B.

Next

assume

that

p

g

q

and

that

R

-LRB-

s

-RRB-

contains

more

than

one

nonsingular

p?p

submatrix

Fundamental

in

the

present

paper

is

now

the

following

property

The

greatest

common

divisor

of

the

p-th

order

minors

of

R

-LRB-

s

-RRB-

generically

is

a

monomial

in

s

-LRB-

a

single

power

of

s

-RRB-

-LRB-

cf.

-LSB-

-RSB-

proposition

-RRB-

In

other

words

generically

the

p-th

order

minors

only

have

zeros

in

common

at

s

The

multiplicity

of

the

zero

at

s

of

a

p-th

order

minor

equals

its

trailing

degree

This

degree

can

be

obtained

as

the

minimum

among

the

ffi

weights

of

the

p-th

order

matchings

in

the

subgraph

corresponding

to

the

square

submatrix

associated

to

the

minor

-LRB-

see

theorem

-RRB-

The

degree

of

the

greatest

common

divisor

can

then

be

computed

by

minimization

Theorem

Let

p

g

q

and

let

R

-LRB-

s

-RRB-

contain

more

than

one

nonsingular

p

p

submatrix

The

greatest

common

divisor

of

all

the

p-th

order

minors

of

R

-LRB-

s

-RRB-

generically

has

a

leading

degree

equal

to

the

minimum

among

the

ffi

weights

of

all

maximum

order

matchings

in

B.

For

proofs

of

the

above

theorems

we

refer

to

-LSB-

-RSB-

There

it

is

also

indicated

how

a

matching

of

maximum

order

can

efficiently

be

computed

Given

a

maximum

order

matching

the

bipartite

graph

B

can

be

decomposed

into

three

different

interconnected

subgraphs

This

decomposition

is

the

so-called

Dulmage

Mendelsohn

decomposition

-LRB-

DM

decomposition

for

short

-RRB-

-LRB-

cf.

-LSB-

-RSB-

-RRB-

The

vertex

sets

V

and

W

are

decomposed

as

V

-LSB-

-LSB-

and

W

-LSB-

-LSB-

with

Vi

Vj

and

Wi

Wj

for

all

i

j

i

j

The

arc

set

A

is

decomposed

as

A

-LSB-

-LSB-

-LSB-

-LSB-

-LSB-

Here

the

set

Aij

contains

all

arcs

in

A

from

vertices

in

Wj

to

vertices

in

Vi

i

j

i

j.

Furthermore

in

general

and

We

have

ignored

here

some

special

cases

For

instance

the

case

in

which

both

and

do

not

appear

together

with

the

arc

sets

and

Except

for

such

special

cases

the

DM

decomposition

has

the

next

general

properties

ffl

For

the

subgraph

defined

by

-LRB-

-RRB-

contains

at

least

two

maximum

matchings

of

order

of

which

the

sets

of

initial

vertices

do

not

coincide

ffl

For

the

subgraph

defined

by

-LRB-

-RRB-

contains

at

least

two

maximum

matchings

of

order

of

which

the

sets

of

terminal

vertices

do

not

coincide

ffl

For

-LRB-

-RRB-

the

subgraph

defined

by

-LRB-

-RRB-

contains

at

least

one

maximum

matching

of

order

The

above

decomposition

is

a

rough

version

of

the

DM

decomposition

The

full

DM

decompostion

is

more

refined

but

this

refinement

is

not

relevant

for

the

present

theoretical

development

In

the

actual

computations

the

refinement

may

be

profitable

In

-LSB-

-RSB-

an

algorithm

for

computing

a

-LRB-

refined

-RRB-

DM

decomposition

is

given

together

with

some

proofs

of

the

above

properties

Recall

that

the

graph

B

is

constructed

starting

from

the

polynomial

matrix

R

-LRB-

s

-RRB-

The

above

introduced

DM

decomposition

for

B

implies

that

R

-LRB-

s

-RRB-

after

some

suitable

row

and/or

column

permutations

can

be

depicted

as

below

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

The

matrix

RViWj

-LRB-

s

-RRB-

has

dimensions

jVij

jWjj

i

j

i

j

The

's

denote

zero

matrices

of

suitable

dimensions

If

the

first

block

row

does

not

appear

Similar

statements

can

be

given

in

case

other

sets

consist

of

zero

elements

To

treat

the

most

general

case

we

assume

in

the

following

that

From

the

theorems

above

it

follows

that

generically

rank

-LRB-

s

-RRB-

rank

-LRB-

s

-RRB-

rank

-LRB-

s

-RRB-

Hence

generically

-LRB-

s

-RRB-

has

full

row

rank

-LRB-

s

-RRB-

is

invertible

and

-LRB-

s

-RRB-

has

full

column

rank

Furthermore

-LRB-

s

-RRB-

contains

at

least

two

distinct

nonzero

minors

of

order

and

-LRB-

s

-RRB-

contains

at

least

two

distinct

nonzero

minors

of

order

We

have

denoted

p

for

the

generic

rank

of

R

-LRB-

s

-RRB-

implying

that

we

can

write

p

where

Now

let

R

-LRB-

s

-RRB-

have

the

above

decomposition

and

let

the

index

set

J

W

jJ

j

p

be

such

that

RVJ

-LRB-

s

-RRB-

generically

has

full

column

rank

Then

J

-LSB-

-LSB-

for

some

Hence

RVJ

-LRB-

s

-RRB-

B

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

CA

So

any

generically

full

column

rank

submatrix

RVJ

-LRB-

s

-RRB-

of

R

-LRB-

s

-RRB-

consisting

of

p

columns

contains

all

the

columns

of

R

-LRB-

s

-RRB-

with

index

in

-LSB-

together

with

columns

of

R

-LRB-

s

-RRB-

with

index

in

Next

for

such

J

it

is

easy

to

see

that

any

p

p

submatrix

of

RVJ

-LRB-

s

-RRB-

that

generically

has

full

rank

contains

all

the

rows

of

RVJ

-LRB-

s

-RRB-

with

index

in

-LSB-

together

with

rows

of

RVJ

-LRB-

s

-RRB-

with

index

in

Hence

the

p

p

submatrices

of

RVJ

-LRB-

s

-RRB-

that

generically

are

nonsingular

only

differ

from

each

other

in

the

last

rows

with

index

in

By

the

remarks

at

the

end

of

the

previous

section

it

is

now

immediate

that

det

-LRB-

grd

-LRB-

RVJ

-LRB-

s

-RRB-

-RRB-

-RRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

det

-LRB-

grd

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

According

to

-LSB-

-RSB-

-LRB-

proposition

-RRB-

the

order

minors

of

-LRB-

s

-RRB-

-LRB-

there

are

at

least

two

-RRB-

generically

only

have

zeros

in

common

at

s

The

multiplicity

of

these

common

zeros

is

given

by

deg

-LRB-

det

-LRB-

grd

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

The

latter

can

be

computed

as

the

minimum

of

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

taken

over

all

where

Hence

deg

-LRB-

det

-LRB-

grd

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

min

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

and

deg

-LRB-

det

-LRB-

grd

-LRB-

RVJ

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

min

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

Note

that

the

last

two

terms

in

the

above

expression

are

independent

of

the

subset

Therefore

maximizing

deg

-LRB-

det

-LRB-

grd

-LRB-

RVJ

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

with

respect

to

J

Wp

as

required

in

section

amounts

to

maximizing

deg

det

-LRB-

-LRB-

s

-RRB-

-RRB-

with

respect

to

where

Combining

the

above

observations

our

main

result

follows

Theorem

a

-RRB-

For

T

Z

the

generic

dimension

of

a

minimal

realization

is

given

by

max

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

min

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

b

-RRB-

For

T

Z

the

generic

dimension

of

a

minimal

realization

is

given

by

-LRB-

max

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

min

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

-LRB-

min

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

max

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

-RRB-

Next

denote

n

i

for

the

maximum

ffi

weight

of

a

maximum

order

matching

and

n?i

for

the

minimum

ffi

weight

of

a

maximum

order

matching

in

the

bipartite

graph

Bi

-LRB-

Vi

Wi

Aii

-RRB-

i

Then

it

follows

directly

from

theorem

that

n

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

n

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

Generalizing

theorem

to

also

hold

for

full

row

rank

or

full

column

rank

matrices

it

follows

that

n

max

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

n

min

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

and

n

max

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

n

min

deg

-LRB-

det

-LRB-

-LRB-

s

-RRB-

-RRB-

-RRB-

With

these

observations

and

notations

we

obtain

the

next

graph

theoretic

version

of

theorem

Corollary

b

-RRB-

For

T

Z

the

generic

dimension

of

a

minimal

realization

is

given

by

n

n

n

b

-RRB-

For

T

Z

the

generic

dimension

of

a

minimal

realization

is

given

by

-LRB-

n

n

n

-RRB-

-LRB-

n

n

n

-RRB-

The

above

corollary

implies

that

the

generic

dimension

of

a

minimal

realization

can

be

obtained

by

computing

maximum

ffi

weights

and

minimum

ffi

weights

of

maximum

order

matchings

This

can

be

done

by

means

of

efficient

graph

theoretic

methods

-LRB-

cf.

-LSB-

-RSB-

-RRB-

Worked

example

In

this

section

we

illustrate

the

results

obtained

in

the

previous

section

Let

R

-LRB-

s

-RRB-

be

given

as

follows

R

-LRB-

s

-RRB-

BBBBBBB

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

OE

-LRB-

s

-RRB-

ffl

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

CCCCCCCA

Assume

that

the

coefficients

in

the

entries

of

R

-LRB-

s

-RRB-

are

unknown

but

that

it

is

known

for

each

entry

what

the

highest

and

lowest

power

of

s

are

BBBBBBB

CCCCCCCA

BBBBBBB

CCCCCCCA

The

numbers

in

and

denote

the

exponent

of

the

highest

and

the

lowest

power

of

s

respectively

present

in

the

corresponding

entry

of

R

-LRB-

s

-RRB-

For

instance

the

highest

power

of

s

present

in

the

-LRB-

-RRB-

entry

-LRB-

s

-RRB-

has

exponent

and

the

lowest

power

has

exponent

i.e.

deg

-LRB-

-LRB-

s

-RRB-

-RRB-

and

deg

-LRB-

-LRB-

s

-RRB-

-RRB-

The

dot

denotes

that

the

corresponding

entry

of

R

-LRB-

s

-RRB-

is

identically

equal

to

zero

and

therefore

does

not

contain

a

highest

or

lowest

power

The

above

numbers

will

be

used

below

as

arc

weights

The

structure

of

the

matrix

R

-LRB-

s

-RRB-

can

be

represented

by

the

bipartite

graph

B

-LRB-

V

W

A

-RRB-

The

set

of

row

indices

V

and

the

set

of

column

indices

W

The

set

A

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

g.

From

the

matrices

and

the

weights

of

the

arcs

can

be

determined

For

instance

the

arc

-LRB-

-RRB-

has

ffi

weight

and

ffi

weight

The

latter

follows

from

ffi

-LRB-

-RRB-

and

ffi

-LRB-

-RRB-

The

graph

B

-LRB-

V

W

A

-RRB-

can

be

depicted

as

below

In

the

picture

already

the

DM

decomposition

has

been

incorporated

-LRB-

see

also

later

-RRB-

This

is

done

merely

for

reasons

of

convenience

In

general

the

DM

decomposition

is

not

known

a

priori

and

an

algorithm

like

in

-LSB-

-RSB-

has

to

be

used

to

compute

the

decomposition

BBBBBBB

BBBBBBB

BBBBBBB

BBBBBBB

BBBBBBN

BBBBBBN

ss

BBBBBBN

AE

BBBBBBN

V

q

q

q

q

q

q

q

q

q

q

q

q

q

In

the

above

picture

it

is

seen

that

any

maximum

order

matching

has

order

Hence

from

theorem

it

follows

that

generically

the

rank

of

R

-LRB-

s

-RRB-

will

be

As

explained

before

the

graph

in

the

above

picture

is

already

in

the

special

form

of

the

DM

decomposition

This

fact

is

emphasized

even

more

in

the

next

picture

BBBBBBB

BBBBBBB

BBBBBBB

BBBBBBB

BBBBBBN

BBBBBBN

ss

BBBBBBN

AE

BBBBBBN

q

q

q

q

q

q

q

q

q

q

q

q

q

Clearly

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

g

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

g

f

-LRB-

-RRB-

-LRB-

-RRB-

g

f

-LRB-

-RRB-

g

f

-LRB-

-RRB-

g

f

-LRB-

-RRB-

g

Now

consider

each

of

the

bipartite

subgraphs

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

individually

It

is

clear

that

and

contain

maximum

matchings

of

orders

and

respectively

More

in

detail

contains

matchings

of

order

contains

matchings

of

order

and

contains

matchings

of

order

For

instance

f

-LRB-

-RRB-

-LRB-

-RRB-

g

and

f

-LRB-

-RRB-

-LRB-

-RRB-

g

are

two

maximum

order

matchings

in

According

and

the

ffi

weights

of

f

-LRB-

-RRB-

-LRB-

-RRB-

g

and

f

-LRB-

-RRB-

-LRB-

-RRB-

g

are

and

respectively

and

the

ffi

weights

of

f

-LRB-

-RRB-

-LRB-

-RRB-

g

and

f

-LRB-

-RRB-

-LRB-

-RRB-

g

are

and

respectively

The

weights

of

the

other

matchings

can

be

determined

similarly

As

before

we

denote

n

i

for

the

maximum

ffi

weight

of

a

maximum

order

matching

and

n?i

for

the

minimum

ffi

weight

of

a

maximum

order

matching

in

the

bipartite

graph

Bi

i

Then

it

follows

that

n

n

n

and

n

n

n

From

our

main

result

it

now

follows

that

if

the

matrix

R

-LRB-

s

-RRB-

is

used

in

an

AR-description

with

time

axis

T

Z

then

the

transfer

matrix

corresponding

to

a

causal

splitting

of

the

external

variable

generically

can

be

realized

by

means

of

a

minimal

realization

with

dimension

n

n

n

If

the

matrix

R

-LRB-

s

-RRB-

is

used

in

an

AR-description

with

time

axis

T

Z

then

a

minimal

realization

of

this

transfer

matrix

generically

has

dimension

-LRB-

n

n

n

-RRB-

-LRB-

n

n

n

-RRB-

Remarks

and

conclusions

In

this

paper

we

presented

a

graph

theoretic

method

for

computing

the

generic

dimension

of

the

state

space

of

a

minimal

realization

The

realization

models

the

relation

between

the

input

and

the

output

obtained

from

a

causal

splitting

of

the

external

variable

in

a

so-called

AR-description

with

the

polynomial

AR-matrix

R

-LRB-

s

-RRB-

In

this

paper

it

is

only

required

to

know

which

are

the

exponents

of

the

highest

and

lowest

powers

of

s

for

each

nonzero

entry

of

R

-LRB-

s

-RRB-

The

methods

developed

are

based

on

well-known

and

efficient

methods

from

graph

theory

and

combinatorial

optimization

References

-LSB-

-RSB-

J.C.

Willems

From

time

series

to

linear

system

Part

I

Finite

dimensional

linear

time

invariant

systems

Automatica

Vol

pp.

Part

II

Exact

modelling

Automatica

Vol

pp.

Part

III

Approximate

modelling

Automatica

Vol

pp.

-LSB-

-RSB-

J.C.

Willems

Models

for

dynamics

Dynamics

Reported

Vol

pp.

-LSB-

-RSB-

J.C.

Willems

Paradigms

and

puzzles

in

the

theory

of

dynamical

systems

IEEE

Trans

Automat

Contr.

Vol

pp.

-LSB-

-RSB-

J.W.

van

der

Woude

Causality

relations

in

linear

structured

systems

Communications

Control

and

Signal

Processing

Proc

Bilkent

Intern

Conference

on

New

Trends

in

Communication

Control

and

Signal

Processing

E.

Arikan

ed.

Elsevier

Amsterdam

pp

-LSB-

-RSB-

K.

Murota

Systems

Analysis

by

Graphs

and

Matroids

Structural

Solvability

and

Controllability

Springer

Verlag

Berlin

-LSB-

-RSB-

W.A.

Wolovich

Linear

Multivariable

Systems

Springer

Verlag

New

York

-LSB-

-RSB-

F.R.

Gantmacher

The

Theory

of

Matrices

Chelsea

New

York

greatest divisor
row reduced matrices
weighted graphs


R?sum ? La vid?oconf?rence sur ordinateur est devenue une r?alit ? : aujourd ` hui on peut recevoir ? partir de son micro-ordinateur de bureau un s?minaire ? distance , participer ? une t?l ? - r?union , etc. . Les logiciels de vid?oconf?rence sur l'Internet sont d'ailleurs devenus d'usage courant . La contribution de nos travaux porte sur deux points principaux . Le premier est de montrer que la technologie actuelle permet de r?aliser des codeurs-d ? codeurs -LRB- codecs -RRB- vid?o en logiciel et donc ? moindre co?t . Le deuxi?me point est de montrer qu'il est possible de transmettre de la vid?o de bonne qualit ? sur l'Internet actuel , et donc en g?n?ral sur des r?seaux sans r?servation de ressources . Ces travaux ont ? t ? illustr?s gr?ce au logiciel de vid?oconf?rence IVS -LRB- INRIA Videoconferencing System -RRB- que nous avons developp ? . Ce logiciel incorpore un codec vid?o qui suit le standard de compression H. 261 de l'Union Internationale des T?l?communications -LRB- UIT -RRB- . Ce standard a ? t ? ? l'origine con?u pour ? tre utilis ? sur des r?seaux ? commutation de circuits de type R?seau Num?rique ? Int?gration de Service -LRB- RNIS -RRB- . Cette th?se d?crit les algorithmes que nous avons ? labor?s et mis en ? uvre pour pouvoir ? mettre eOEcacement de la vid?o sur des r?seaux ? commutation de paquets comme l'Internet . Plus particuli?rement , nous proposons un algorithme de d?coupage en paquets du AEot de bits H. 261 , des algorithmes de contr?le d'erreurs contre la perte des paquets , des algorithmes de contr?le de d?bit pour codeur H. 261 et enon un algorithme de contr?le de congestion qui adapte le d?bit du codeur ? la bande passante disponible du r?seau . Enon , nous discutons du probl?me g?n?ral de la transmission vid?o vers un ensemble h?t?rog?ne de r?cepteurs et proposons des solutions . Certaines d'entre elles sont applicables dans l'Internet actuel et d'autres ne le seront qu ' ? moyen terme avec les changements escompt?s dans l'architecture de l'Internet . Mots cl?s : Codage en sous-bandes , Compression vid?o , Contr?le de congestion , H. 261 , Internet , MBone , Multim?dia , Vid?oconf?rence . Summary Multimedia applications have now become mainstream applications over networks such as the Internet . Examples of such applications include videoconferencing , remote teaching , etc. . Our work and contribution has focused on two main points . First , we have shown that the current generation of workstations is powerful enough to support real-time video software codecs . Second , we have shown that it is possible to transmit good quality video over the Internet , and more generally over networks which do not provide resource guarantees . This work has been illustrated in our IVS -LRB- INRIA Videoconferencing System -RRB- software . This software includes a video codec which implements the H. 261 video compression standard from the International Telecommunication Union -LRB- ITU -RRB- . This standard was originally designed for circuit switched networks such as the Integrated Services Digital Network -LRB- ISDN -RRB- . This thesis describes algorithms we developed to send live video over packet switched networks such as the Internet . Speciocally , we propose a packetization scheme of H. 261 video data , error control schemes to prevent and compensate for packet loss , video rate control algorithms and a congestion control algorithm that adjusts the output rate of the video coder depending on network conditions . We also consider issues related to multicast video distribution in heterogeneous packet networks and propose some solutions . Some of them can be used on the today 's Internet while others are more suited to the emerging new Internet architecture . 261 , Internet , MBone , Multimedia , Subband coding , Videoconferencing .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O