The Dimension of Matrices Matrix Pencils with Given Jordan

Kronecker Canonical Forms

James W Demmel

y

Alan Edelman

Novemb er

Abstract

The set of n by n matrices with a given Jordan canonical form denes a subset of matrices

2

in complex n dimensional space We analyze one classical approach and one new approach to

count the dimension of this set The new approach is based up on and meant to give insight

into the staircase algorithm for the computation of the Jordan Canonical Form as well as the

o ccasional failures of this algorithm We extend b oth techniques to count the dimension of

the more complicated set dened by the Kronecker canonical form of an arbitrary rectangular

matrix p encil A B

Intro duction

Given any square matrix A the set of matrices similar to A forms a manifold in complex n

dimensional space This manifold is of course the orbit of A under the action of conjugation

orbitA fP AP detP g

The matrix p encil analog is to consider any pair of m by n matrices A and B and dene the

orbit of the matrix p encil A B by the action of multiplication on the left and right by square

nonsingular matrices of the appropriate size

orbitA B fP A B Q detP detQ g

This orbit denes a manifold of p encils in mn dimensional space All p encils on this manifold are

said to b e equivalent to A B In matrix theory a p encil refers to a linear matrix p olynomial

often in the indeterminate See

Our concern in this work is to count the codimension of these manifolds as ob jects in complex

Euclidean space For simplicity of exp osition we sometimes refer to these two problems as counting

the codimension of a single matrix or of a matrix p encil when more prop erly we would refer to

counting the codimension of the orbits We take two approaches one based on classical techniques

Computer Science Division and Department of Mathematics University of California Berkeley CA dem

melcsb erkeleyedu Supp orted by NSF grants DMS ASC and under a sub contract with the

University of Tennessee under DARPA contract DAALC

y

Department of Mathematics Ro om Massachusetts Institute of Technology Cambridge MA edel

manmathmitedu Supp orted by NSF grant DMS and the Applied Mathematical Sciences subprogram of

the Oce of Energy Research US Department of Energy under Contract DEACSF

that identify the tangent spaces of these manifolds and the other based up on existing numerical

algorithms for computing the Jordan and Kronecker forms

The classical approach to solving this problem requires the computation of the tangent space

to the orbits In the single matrix case the tangent vectors have the form

XA AX

while in the matrix p encil case the tangents have the form

X A B A B Y

Thus the co dimension of the single matrix orbit is the numb er of linearly indep endent matrices X

for which vanishes while the co dimension of the matrix p encil orbit is related to the numb er

of linearly indep endent matrix pairs X Y for which vanishes

Arnold has rederived the formula for the Jordan case for the purp ose of dening a particular

normal form for deformations of a matrix with a given Jordan form This form is convenient

b ecause of its minimum numb er of parameters We are unaware of any general dimension count

for matrix p encils in the literature One partial result of Waterhouse counts the co dimension

of a singular pair of n by n matrices ie the square case to b e n

Our new approach is based on the so called staircase algorithms for the Jordan and Kronecker

canonical forms The staircase algorithm for the Jordan canonical form pro ceeds by computing the

Weyr characteristics of the matrix while the staircase canonical form pro ceeds by computing a

more complicated set of structural indices

In this pap er we lay the groundwork for a theory that we hop e might explain the o ccasional

failures of existing staircase algorithms to nd the right Jordan or Kronecker form These al

gorithms are used in systems and control theory to nd the input matrix or p encil of highest

co dimension within a usersupplied distance of the input data The structures of these matrices

or p encils reect imp ortant physical prop erties of the systems they mo del such as controllability

The user cho oses to measure the uncertainty in the data The existence of a matrix or

p encil with a dierent structure within distance of the input means that the actual system may

have a dierent structure than the approximation supplied as input So the goal of these algorithms

is to p erturb the input by at most so as to nd the matrix or p encil of as high a co dimension as

p ossible The algorithm is said to fail if there is another p erturbation of size at most which would

raise the co dimension even further Therefore we need to understand how the algorithm pro duces

outputs of each co dimension which is explained in this pap er although this is just a rst step to

explaining the failures In particular this is why we need to prove a known result Theorem

using a new technique staircase form We b elieve the dimension count for the matrix p encil case

Theorem is new

Main Results

Theorem The codimension of the orbit of a given matrix A is

X

q q q c

Jor

where q q q denotes the sizes of the Jordan blocks of A corresponding to

Theorem The codimension of the orbit of A B depends only on its Kronecker structure

This codimension can be computed as the sum of separate codimensions as given in the table below

This equation is expressed more compactly in Equation in the next section Section provides

examples of how to use these formulas Readers already familiar with the Kronecker form may wish

to pro ceed directly to Section b efore reading the pro ofs

Breakdown of the codimension count

The co dimension of the orbit of A B dep ends only on its Kronecker structure It can b e

computed as the sum c c c c c c whose comp onents are

Total Jor Right Left JorSing Sing

dened as

The co dimension of the Jordan structure

X

c q q q

Jor

where the sum is over all eigenvalues as in Theorem including the innite eigenvalue

if it is present

The co dimension of the L singular blo cks

X

c j k

Right

j k

where the sum is taken over all pairs of blo cks L and L for which j k

j k

T

The co dimension of the L singular blo cks

X

c j k

Left

j k

T T

and L for which j k where the sum is taken over all pairs of blo cks L

j

k

The co dimensions due to interactions of the Jordan structure with the singular blo cks

c size of Jordan structurenumb er of singular blo cks

JorSing

Here the numb er of singular blo cks counts b oth the left and the right blo cks

T

The co dimensions due to interactions b etween L and L singular blo cks

X

c j k

Sing

jk

T

where the sum is taken over all pairs of blo cks L and L

j

k

These are complex co dimensions but the answers are correct for real co dimensions when the

matrices or matrix p encils have real Jordan or Kronecker forms For the rest of this pap er all

dimensions will b e complex dimensions half the numb er of real dimensions

Mathematical Preliminaries and Notation

Matrix Canonical Forms

The basic notation in this area has b een reinvented by many authors So as to make this work

selfcontained and also to x notation we review the basic denitions Further information may

b e found in standard matrix theory texts such as or

Given a matrix A that has only one eigenvalue it is always p ossible to nd a similarity that

transforms A into the form

J A diagJ J

q q

1 2

where J is a q by q matrix with on the diagonal and on the sup erdiagonal known as a Jordan

q

blo ck

For an arbitrary matrix it is always p ossible to nd a similarity that transforms A into a union

of blo cks of the form

1 2

J A diagJ A J A

where denotes the distinct eigenvalues of A

To x the order of the Jordan blo cks within we assume

q q

but we do not x the order of the eigenvalues

Denition The matrix J A dened up to eigenvalue orderings is known as the Jordan

Canonical Form of A

Denition The sequence of numbers q dened above gives the sizes of the Jordan blocks

i

for the eigenvalue They are known as the Segre characteristics of A relative to

It is sometimes convenient to think of this as an innite sequence with q for j the

j

numb er of Jordan blo cks corresp onding to

q

i

Denition The elementary divisors of the matrix A xI are the polynomials x in

the indeterminate x where is an eigenvalue of A and q is a Segre characteristic corresponding

i

to

Q

Denition The invariant factors of the matrix A xI are the polynomials P x

i

q

i

x It fol lows that if we let p denote the degree of the ith invariant factor then

i

X

q p

i i

P

Of course n p b ecause this counts the sizes of all the Jordan blo cks of all the eigenvalues

i

of A

Some authors see pages and consider the quantity m dened as the degree of the

i

greatest common divisor of all the i by i minors of the linear matrix p olynomial A I It can b e

shown that m p p

i ni n

Denition The nullity of an n by n matrix A is n rankA For m by n matrices the row

nul lity and the column nul lity are m rankA and n rankA respectively

Denition Let w denote the dierence

j

j j j j

nullity A I nullity A I rankA I rankA I

The numbers w are the Weyr characteristics of A relative to

j

It turns out that the numb er of blo cks J with q j is exactly w The dimension of the

j

q

nullspace of A I is w

The following lemma is critical for the construction of the staircase algorithm

Lemma Let Q be any unitary matrix whose rst w columns span the nul lspace of A I

Then

w n w

I S

T

Q AQ

A

where A is an n w by n w matrix With the deletion of w the Weyr characteristics of

A are the same as that of A In particular the Weyr characteristics of the other eigenvalues are

unchanged

Pro of Idea

T

Without loss of generality we may assume that Q AQ is already a row and column p ermutation

of a Jordan form The Jordan structure of A is the same as the Jordan structure of A except that

every Jordan blo ck of A corresp onding to the eigenvalue is exactly one dimension smaller

Let A B b e an m by n matrix p encil When discussing the Kronecker case is always an

indeterminate It is p ossible to nd an equivalent p encil KronA B in the Kronecker Form

T T

J J L KronA B diagL L L

g

1

1

h

The L blo cks are by rectangular blo cks with on the diagonal and on the sup erdiagonal

T

The L blo cks are by with on the diagonal and on the sub diagonal The and

sometimes referred to as the size can b e leading to columns and rows resp ectively The J

blo ck is of the form with the addition of I This constitutes the Jordan structure of the nite

eigenvalues The J blo ck is the union of blo cks of size q each of which has on the main

i

diagonal and on the sup erdiagonal This constitutes the Jordan structure corresp onding to the

innite eigenvalue Frequently there will b e no need to distinguish b etween the nite and innite

eigenvalues Indeed with an appropriate Mobius transformation sending A B to A B

A B all eigenvalues may b e assumed nite

T

The L and L blo cks constitute the singular part of the p encil The Jordan structure for nite

and innite eigenvalues constitutes the regular part of the p encil The Segre characteristics remain

well dened for a matrix p encil but we must include the characteristics for the innite eigenvalue

as well

Denition Let

g

denote the sizes of the g L blocks of a pencil and let

h

T

denote the sizes of the h L blocks Then the numbers are known as the column minimal

i

indices while the are the row minimal indices

i

We can now recast Theorem using the notation from the previous denitions The co di

mension of the orbit of A B can b e written compactly as

X

co dorbitA B p p p g h p

i

X X X

i j i j i j

ij

i j i j

where the p include any innite eigenvalue blo cks

i

Conjugate Partitions

The Weyr characterists and the Segre characterists of a matrix for a given eigenvalue are closely

related

Denition Let k k k be a partition of the positive integer k ie k

k k Let l denote the number of k that are greater than or equal to j Then the l form

j i j

a partition of k known as the conjugate partition of the k

i

It is easy to verify that the prop erty of b eing a conjugate partition is symmetric For example

are conjugate partitions of This is easy to verify by reading

the diagram b elow known as a Ferrers diagram vertically and horizontally

The idea of the conjugate partition is very simple yet very p owerful It allows the interchange

of summations

l

j k

i

X X X X

f i j f i j

i j j i

where f i j is any function of i and j and the k and l are conjugate partitions

i j

Lemma The Weyr characteristics and the Segre characteristics of a matrix corresponding to

a particular eigenvalue are conjugate partitions

The pro of of this lemma is evident from the Jordan form of the matrix p

A Fundamental Co dimension Count

Our co dimension counts for the Jordan and Kronecker form are built up from the fundamental

Lemma To state it we need to intro duce a little notation from manifold theory

Denition The set of k dimensional subspaces of n dimensional space along with its natural

manifold structure forms the Grassmann manifold denoted G n

k

The Grassman manifold and its dual G n are isomorphic of dimension k n k In

nk

Lemma we will need a fullrank parameterization for G n which we construct as

nk

follows Recall that a chart for a complex ddimensional manifold M is an op en neighb orho o d U

d

in C plus a homeomorphism from M into U A full rank parameterization is the inverse of this

homeomorphism Because of the action of the unitary group it suces to sp ecify a lo cal full rank

parameterization near any one element say E the one generated by the rst k co ordinate vectors

k

We create a parameterization from unitary matrices of the form

I R R I R

Q

R I I RR

where R is n k by k The homeomorphism maps complex n k by k matrices R to the span of

nk k

the rst k columns of Q If Q is any xed unitary matrix the homeomorphism from R C

to the space spanned by the rst k columns of QQ provides the parameterization mapping from

nk k

a neighb orho o d of the origin in C to a neighb orho o d in G n of the space spanned by the

k

rst k columns of Q

Lemma The set of m by n matrices with rank r is a manifold with codimension m r n r

Pro of We construct a parameterization whose image is a neighb orho o d of a particular m by n rank

r nr mr

r matrix A as follows A neighb orho o d of the origin in the pro duct space C C will serve

as a domain for the parameterization Let Q b e any unitary matrix whose rst n r columns span

the nullspace of A so that AQ M is zero in its rst n r columns and its last r columns M

r nr mr

have full rank Let Q b e as in with k n r Then the map from R T C C

to M T Q Q is the desired homeomorphism If m n then we may equally well use the

Q Thus the dimension is r n r mr homeomorphism mapping R T to QQ M T Q

and the co dimension is mn r n r mr m r n r

We graphically depict the indep endent parameters as follows

r

n r

m r

S

r

R

A

T T T

Here R refers to the co ordinates that dene the null space while T S A is the matrix

mr

in C The black square in the upp er left clearly indicates the co dimension of m r n r

Later we will take advantage of this construction to recursively construct further submanifolds

by placing analogous rank constraints on A so that A still lies in a small neighb orho o d of the

origin Therefore it will b e easy to see that we need merely add the co dimensions of our constraints

at each level in order to compute the overall co dimension of the nal submanifold Indeed the

parameterization of Lemma is constructed explicitly at each step of the staircase algorithm

Pro ofs of Theorem Co dimension Count for Jordan Form

Classical Pro of

Consider conjugating the matrix A by I X where is a small scalar This yields

I X AI X A XA AX O

from which it is evident that the tangent space to orbitA at A consists of the matrices of the form

XA AX The dimension of the orbit is equal to the dimension of the tangent space so that the

co dimension of the orbit is equal to the dimension of the nullspace of the mapping that sends X

to XA AX The co dimension of the orbit is then the numb er of linearly indep endent solutions

to AX XA This numb er of solutions is well known to b e

p p p

See page of volume of

An alternative expression for the numb er of solutions to AX XA is

n m m

n

as given in According to the remark following Denition these expressions are identical

Outline of the Staircase Algorithm

The staircase algorithm for the computation of the Jordan Canonical Form app ears in

Some references refer to stairacase form to mean a slightly dierent concept The

staircase algorithm of interest to us computes the Weyr characteristics It is built recursively up on

the idea in Lemma

Staircase algorithm for computing the Jordan form for eigenvalue

i

A A I

tmp

while A not full rank

tmp

i i

P

i

w and n n n dimA Let n

j tmp tmp

j

Compute an n by n unitary matrix Q whose leading w columns span the null space

tmp tmp i

of A

tmp

Q Q A diag I A diag I

n n

Let A b e the lower right n w by n w corner of A

tmp tmp i tmp i

A A I

tmp tmp

endwhile

The nal A is easily seen to b e unitarily similar to the initial A The nal A is in staircase

form as illustrated with the following example

w w w w n

w I A

w I A

w I A

w I

n A

Here the sup erdiagonal blo cks A the stairs and also A I are of full column rank

ii

while the staircase region in the lower triangle is entirely If A has only one eigenvalue then n

is and the last blo ck row and blo ck column do not app ear If A has other eigenvalues then the

staircase form corresp onding to the remaining eigenvalues may b e extracted by applying the same

algorithm to A

An easy observation is that

Lemma The w computed by the staircase algorithm for the eigenvalue are the Weyr char

i

acteristics corresponding to the eigenvalue

Second Pro of of Theorem

Let A b e any matrix We will show that the staircase algorithm in eect creates a parameterization

for an op en neighb orho o d N A of A on the manifold orbitA Let b e an eigenvalue of A

Then A I has rank n w The indep endent parameters p ortrayed in may b e used as a

parameterization for a neighb orho o d of A I on the manifold of rank n w matrices Lemma

tells us that we have a parameterization for orbitA if we make further assumptions on the

Jordan structure of A Notice that in a small enough neighb orho o d of A the last n r columns

of the staircase form are full rank It is imp ortant to observe the indep endence of the w n w

parameters in R from the w n w parameters of S and the as of yet uncounted parameters in

A The rst eigenvalue is fully parameterized when A I has full rank The parameters are

pictorially depicted b elow in an example that recurs two more times b efore A I has full rank

w

S

w

S

w

S

R

R

R

A

w w w

This parameterization pro cess is rep eated on A with a new eigenvalue shift in an identical

manner This rep etition continues until A do es not exist The areas of the black squares in the

gure ab ove indicate the co dimension that we might attribute to the eigenvalue This co dimension

is then

w

i

X X X

w k

i

i i

k

q

k

X X

k

i

k

X

k q

k

k

using the fact that the Weyr and Segre characteristics are conjugate partitions

The total co dimension for the entire Jordan structure of A is obtained by summing over all the

eigenvalues b ecause of the indep endenc e of the parameters

Tangent Space Pro of of Theorem

We include two pro ofs b oth of which we b elieve to b e new The rst pro of requires counting the

numb er of indep endent solutions to two simultaneous matrix equations derived by analyzing the

tangent space while the second pro of in Section requires an analysis of the staircase algorithms

for the Kronecker canonical form

Consider an orbit preserving transformation of the m by n p encil A B obtained by multiplying

on the left by I X and the right by I Y where is a small scalar This yields A B

X A B A B Y O from which it is evident that the tangent space to the orbit

of the p encil consists of the p encils that can b e represented in the form

f X Y X A B A B Y

where X is an m by m matrix and Y is an n by n matrix

Since maps a space of dimension m n linearly into a space of dimension mn the

dimension of the image space is m n d where d is the dimension of the kernel of f X Y

and so the co dimension is

mn m n d d m n

The term m n represents extra baggage due to our consideration of rectangular p encils As in

the Jordan case we need to calculate d the numb er of linearly indep endent solutions to f X Y

This can b e written as the two simultaneous equations

XA AY and XB B Y

Unfortunately we can not simply quote a classical count of the numb er of indep endent solutions

to as we were able to do in Section However since

P f X Y Q PXP P A B Q P A B Q QY Q

it follows that the numb er of linearly indep endent solutions to f X Y dep ends only on the

Kronecker structure of A B Thus we assume that A B is already in Kronecker canonical

form M diag M M The Kronecker case is more complicated than the Jordan case due to

the greater numb er of p ossibilities for the Kronecker structure M

We partition the equation XM MY conformally with M diag M M so that

X M M Y where M is m by n X is m by m and Y is n by n

ij j i ij k k k ij i j ij i j

m m n n n n n n

m X X m M m M n Y Y

m X X m M m M n Y Y

The next lemma allows us to compute the quantity d mentioned b efore Equation as the

sum of the numb er d of indep endent solutions of X M M Y in the variables X and Y

ij ij j i ij ij ij

Lemma In terms of the above notation

X

d d

ij

ij

Pro of As is evident from the example

X X M M Y Y

X X M M Y Y

the equations X M M Y i j are all mutually indep endent

ij j i ij

Given any two blo cks M and M we allow i j here we dene their interaction and the

i j

cointeraction

Denition Let M be m n and let M be m n Let X be an arbitrary m m matrix and

i i i j j j j i

Y be an arbitrary n n matrix We dene the interaction d of M with M as the dimension

j i ij i j

of the linear space fX Y g such that XM M Y We dene the cointeraction of M with M as

j i i j

c d m n m n We also consider the combined cointeraction which we dene

ij ij i i j j

as c c when i j and simply c when i j

ij j i ii

Notice that the combined cointeraction has a dierent denition dep ending on whether M and

i

M are distinct blo cks even if they happ en to b e equal on one hand or if i j on the other hand

j

Strictly sp eaking the combined cointeraction is a function of M M and the Kronecker delta

i j ij

Lemma The codimension of a matrix pencil M with Kronecker structure diag M M is

the sum of cointeractions of M with M for al l combinations of i and j

i j

Pro of The sum of the cointeractions is

X

fd m n m n g d m n

ij i i j j

ij

as in Equation

We must now count the numb er of linearly indep endent solutions and the asso ciated combined

cointeractions to the following equations

T T

XL L Y and XL L Y

j k

j

k

T T

XL L Y and XL L Y

j k

j

k

XJ L Y and XL JY and related structures

j j

XJ JY

where J denotes the nonsingular structure of the p encil

T T

XL L Y and XL L Y

j k

j k

Consider the equation XL L Y where X is an unknown k by j matrix and Y is an unknown

j k

k by j matrix This equation is equivalent to the two equations

X I I Y

j k

X I I Y

j k

where denotes a column of zeros These two equations are in turn equivalent to the conditions

X Y k j

Y Y k j

Y Y k

j

If j k there is only the trivial solution X and Y The interaction is so that the

cointeraction is j j k k

If j k then there are nontrivial solutions Y can b e any upp er triangular To eplitz matrix

with j k diagonals starting from the main diagonal X is then obtained from Y by omitting

the rst row and column The interaction of L with L is j k so that the cointeraction is

j k

j k j k

We conclude that the combined cointeraction of L and L is while if j k then the combined

j j

cointeraction of L with L is j k

j k

Taking the transp ose and interchanging the roles of j and k we see that the same result holds

T

for blo cks of the form L We also remark that the analysis is correct even if j or k is

j

T T

L Y Y and XL XL L

k j

j k

T

We pro ceed in a manner similar to the previous case Consider the equation XL L Y where

j

k

X is an unknown k by j matrix and Y is an unknown k by j matrix The equations are

equivalent to

X Y k j

Y Y k j

Y Y k

j

X j

k

T

This has only the trivial solution X Y so that the interaction of L with L is and the

j

k

cointeraction is

T T

A similar examination of the equation XL L Y shows that the interaction of L with L is

k k

j j

j k so the cointeraction is j k j k We conclude that the combined cointeraction

is j k

Jordan Blo cks and Singular Blo cks

In one way the computation involving Jordan blo cks is easier since the interaction is equal to the

cointeraction This is true simply b ecause the Jordan blo ck is square However we must now

allow for arbitrary eigenvalues

Assume that J is a single Jordan blo ck of size k corresp onding to the nite eigenvalue e We use

k

e here so that there is no confusion with the indeterminate We consider solutions to XJ L Y

k j

The reader can verify that the dimension of the space of solutions is k Indeed the rst row of

the j by k matrix Y can b e chosen arbitrarily and this determines the remaining elements as

follows Y Y e X is obtained from Y by deleting the last row and eY Y Y

An analogous though simpler argument shows that the case of innite eigenvalues gives the same

answer We can also resort to a Mobius transformation as well We conclude that the interaction

of J with L is k

k j

The interaction of L with J is readily shown to b e From the equation XL J Y we can

j k j k

conclude that X is obtained from Y by deleting the last column that the last column of Y is zero

and if the mth column of Y is then so is the m st column of X and hence so is the m st

column of Y

T T

The cases XL JY and XJ L Y can b e reduced to the previous cases by rememb ering

j j

T

that if J is a Jordan blo ck J PJP where P is the p ermutation that renumb ers indices in

T

backwards order For example the numb er of indep endent solutions to XL JY is the same as

j

T T T

the numb er of solutions to Y P PJ P L X P

j

Jordan Blo cks with other Jordan Blo cks

Let J I b e the entire nonsingular p ortion of the Kronecker structure If we assume that there

are no innite eigenvalues then the equation X J I J I Y implies X Y and then we

are reduced to the case XJ JX in Theorem We remark that Theorem tells us that there

is no interaction among Jordan blo cks with dierent eigenvalues

We omit the tedious algebra but it is p ossible to show that an innite eigenvalue b ehaves

exactly as if it were nite A simpler argument would p oint out that we can rotate the Riemann

sphere to insure that all the eigenvalues are nite without changing the co dimension count We

conclude that the combined cointeractions of the nonsingular p ortion of the p encil is exactly as in

Theorem

Pro of of Theorem

The pro of follows from the analysis of the cases presented in Sections through

Pro of of Theorem Based on the Staircase Algorithm

We b egin by reviewing the staircase algorithm The version we use has three passes Let A B b e

an m by n matrix p encil The rst pass pro duces two sequences of numb ers s and r and returns

i i

a p encil A B with no L blo cks and no zero eigenvalues The sequence satises

j

s r s r s

where

s r the numb er of L blo cks and

i i i

blo cks r s the numb er of J

i i

i

The algorithm is as follows

Staircase algorithm for computing the Kronecker form for the eigenvalue and L blocks

j

i

A A

tmp

while A not full rank

tmp

i i

P

i

Let n s and n n n col sA

j i tmp

j

P

i

Let m r and m m m r ow sA

j i tmp

j

Compute an n by n unitary matrix Q whose leading s nullity A columns span

i i i tmp

the right null space of A

tmp

Q Q and B B diag I Let A A diag I

n n

B B m m n n s

tmp i

Compute an m by m unitary matrix P whose rst r rankB rows span

i i i tmp

the column space of B

tmp

P B P A and B diag I Let A diag I

m m

Let A b e the last m r rows and n s columns of A

tmp i i i i

endwhile

It is easy to see the nal A B is unitarily equivalent to the initial A B We illustrate the

nal form of A B with the following small example

s s s s n

r B A B

r B A B

r B A B

m A B

On completion the B blo cks have full row rank and the A blo cks have full column rank

ii ii

The rst pass through the inner lo op of the algorithm p ostmultiplies A and B by a unitary Q

so As leading s nullity A columns are and then premultiplies A and B by a unitary P so

that B the leading r by s submatrix of B is full rank and the remaining rows of the rst s

columns of B are zero We then rep eat the pro cess on the trailing m r by n s submatrix of

A B to get s and r We continue until the trailing blo ck of A has full rank or is null

Just as with the Jordan form each step of the algorithm incrementally builds a parameterization

for the set of matrices of a given Kronecker form Each step of the algorithm restricts the Kronecker

form of the p encil to a set of higher co dimension The restrictions imp osed at each step are

indep endent for the same reason they were in the Jordan case so we can just add co dimensions

The increase in co dimension at each step is given by Lemma as the sum of the pro ducts of

the row and column nullities of submatrices of A and B Sp ecically the m by n submatrix of

i i

A has column nullity s rank n s row nullity m s n and so by Lemma co dimension

i i i i i i

m s n s Similarly the co dimension due to B at step i is m r s r The rst pass

i i i i i i i i

through the algorithm determines the L and J blo cks so that the co dimension due to these blo cks

is given by

X

fm s n s m r s r g

i i i i i i i i

i

We pro ceed to show that is the formula given in Theorem

For convenience we list our notation

P

i

r m numb er of rows in the lower right subp encil at step i m

k i

k

P

i

s n numb er of columns in the lower right subp encil at step i n

k i

k

s column nullity of A at step i

i tmp

r row rank of B at step i

i tmp

l numb er of L blo cks in the original p encil

i i

T

l numb er of L blo cks in the original p encil

i i

t numb er of J blo cks in the original p encil

i

i

u size of the regular structure corresp onding to

Only left singular blo cks

We b egin by assuming that our p encil only contains left singular blo cks Let l denote the numb er

i

of L blo cks It is easy to show by induction that the algorithm computes

i

X

m j il

i j

j i

X

n j il

i j

j i

X

l s

j i

j i

X

l r

j i

j i

P

To see this rst check that m m and n n Indeed it is obvious that m j l b ecause

j

P

this counts the j rows in each L blo ck It is also obvious that n j l b ecause this counts

j j

the j columns of each L blo ck Just by lo oking at the form of an L blo ck we see that each left

j j

singular blo ck makes a contribution of one to the column nullity of the p encil thus s is the total

numb er of left singular blo cks Finally we have s r l the numb er of L blo cks To check

the validity of the formulas for arbitrary i pro ceed by induction using the denition and prop erties

of m n r and s listed immediately ab ove and at the b eginning of Section

i i i i

When there are only left singular blo cks we see that expression evaluates to

X X

j i l l

j i

j i i

P

from using a dierent notation In our This corresp onds to the term

i j

i j

current notation counts every pair L L for which j i with the weight j i b ecause

i j

there are exactly l l such pairs For example if we have two L blo cks and two L blo cks then

i j

and In the current notation l and l Either way the sum is

four times ie

Left singular blo cks and J blo cks

We now add the assumption that there are J blo cks as well Let t b e the numb er of J blo cks

i

i

ie Jordan blo cks of size i corresp onding to a zero eigenvalue Again by induction it is p ossible to

show

X

j il t m

j j i

j i

X

l n m

j i i

j i

X X

s l t

i j j

j i j i

X

r l t

i j j

j i

Now for this case Expression evaluates to

X X X X X

A A

t l t j i l t l

j j j j j i

i j i j i j i j i

which can readily b e manipulated to b e

X X X X X X

t t l l j i t

j k i j j

i j i j i j i i

k i

where is the same interaction among the left singular blo cks as in Equation We recognize

P

the square of the i st Weyr characteristic of the from Denition that t is w

j

j i

i

P

w is the co dimension zero eigenvalue From our new pro of of Theorem we know that

i i

due to the zero eigenvalue alone

Lastly we must evaluate

X X X X

l t l j i t

j k i j

j i j i i

k i

i

X X X X

t k i t l

k k i

i j

k j k i

i k i

X X X X X X

l t t k i t

i k k k

i j j

k k i k i

X X

k t l

k i

i

k

size of Jordan structure for numb er of left singular blo cks

P

Therefore g q

i

i

Arbitrary singular blo cks and arbitrary Jordan structure

T

We complete the rst pass through the algorithm by dening l to denote the numb er of L blo cks

i i

P

and u to b e the size of the regular Jordan structure for Thus u p q includes the

i

i i

structure for which plays no sp ecial role during the rst pass through the algorithm

We once again omit the details but it is p ossible to show by induction that the algorithm

computes

X

j l u m m

i

j i

j

X

n n j l u

i

i j

j

s s

i

i

r r

i

i

where the sup erscript indicates no right singular structure and no nonzero regular structure ie

as in the notation of Section

We now have that the co dimension expression in is

X X X X X

l l t l j l u

j j i

j j

j j i i j i j

where is as in With some algebraic manipulation we obtain

X X X X

l l i j u l k t l

i i k

j i

ij i i

k

The terms here are the terms

X X X

q p q h g

i i j

i i

i i ij

Second and third passes through algorithm

T

The rst pass through the algorithm gives us a p encil A B which may have only L blo cks

j

T

and nonzero eigenvalues We then run the algorithm on B A so that the indices that gave

the right singular blo cks b efore now give the left singular blo cks The indices that describ ed

now describ e This algorithm returns a p encil with only a regular part that has no zero or

innite eigenvalues

If we reinvoke the previous results we see that the second pass through the algorithms nearly

completes the entire expression The only gap is

X

q q q

fg

This is just the Jordan structure of the regular part other than the zero and innite eigenvalues

This is covered in the third phase of the algorithm completing the pro of

Examples Observations Ab out Genericity and Applications

to the Waterhouse Theorems

We illustrate how these theorems may b e used with a numb er of examples

Let A b e a matrix all of whose eigenvalues are The most generic such matrix whose orbit

has co dimension n is a single Jordan blo ck The least generic such matrix with co dimension

n ie dimension is the single p oint I

P

Let A b e a matrix with no multiple eigenvalues The co dimension of its orbit is then

or n One might intuitively think of this as having sp ecied the n eigenvalues but no other

information ab out the matrix Indeed if you do not wish to sp ecify the value of an eigenvalue

the correct co dimension for this unsp ecied eigenvalue is one less

q q q

In the Kronecker algorithm one sometimes sp ecies that that the eigenvalues are or

other It would therefore b e correct to subtract one for eigenvalues classied as other

Let the Kronecker structure of a particular by p encil b e diagL L L L Since this

p encil has only L blo cks the entire co dimension is to b e found in c It is

j right

Notice that the interactions of two L blo cks that are equal or dier by only one make no

j

contribution to the co dimension If a p encil contains only blo cks of the form L or L the

co dimension is We have therefore observed

Corollary The generic Kronecker structure for a matrix pencil with d n m is

diag L L L L

where bmdc the total number of blocks is d while the number of L blocks is given

by m mo d d which is when d divides m

The same statement holds when d m n if we replace the L and L blo cks by their

transp oses Corollary was obtained by Van Do oren Wilkinson and Wonham as discussed

on page of

T

where j Let an n by n matrix p encil have the Kronecker structure diagL L

j

nj

n From the c p ortion of the co dimension we learn that the orbit has co dimension

Sing

j n j n If a square p encil has any singular part at all it is fairly easy to

check that the smallest p ossible co dimension is n and it must b e of this form We have

thus repro duced a result of Waterhouse

Corollary The generic singular pencils of size n by n have Kronecker structures

T

diag L L

j

nj

where j n

Intuitively we might think of this as the n conditions on the co ecients of that detA

B

T

More generally has shown that if a square matrix has one L blo ck and one L blo ck

r

s

and otherwise has a generic n r s n r s blo ck eigenvalues unsp ecied then

the co dimension is r s n r s n r s This to o readily follows from

our results

T

If an by p encil has the Kronecker form diagL L L J where here J denotes a

single by Jordan blo ck with eigenvalue then c c c

Jor Right JorSing

and c giving a total co dimension of

Sing

T

The p encil has a Kronecker structure consisting of m L blo cks and n L blo cks The

co dimension from c only is mn ie the dimension is

Sing

Acknowledgements

Some of this work was p erformed in Toulouse indeed it was b egun along Rue de Fermat under the

kind hospitality and supp ort of CERFACS We wish to particularly thank Mario Arioli Dominique

Bennett and Iain Du We further would like to thank the referees for their numerous comments

which we feel have b een a great improvement to the pap er

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