A Geometric Approach to Perturbation
Theory of Matrices and Matrix Pencils
Part I Versal Deformations
y y
Alan Edelman Erik Elmroth and Bo K agstr om
March
UMINF
ISSN
Abstract
We derive versal deformations of the Kronecker canonical form by deriving the tan
gent space and orthogonal bases for the normal space to the orbits of strictly equivalent
matrix p encils These deformations reveal the lo cal p erturbation theory of matrix p en
cils related to the Kronecker canonical form We also obtain a new singular value
b ound for the distance to the orbits of less generic p encils The concepts results and
their derivations are mainly expressed in the language of numerical linear algebra We
conclude with exp eriments and applications
Keywords Jordan canonical form Kronecker canonical form generalized Schur decom
p osition staircase algorithm versal deformations tangent and normal spaces singularity
theory p erturbation theory
Department of Mathematics Ro om Massachusetts Institute of Technology Cambridge MA
USA Email address edelman math mit edu Supp orted by NSF grant DMS and an Alfred
P Sloan Foundation Research Fellowship
y
Department of Computing Science Ume a University S Ume a Sweden Email addresses
elmroth cs umu se and bokg cs umu se In part supp orted by the Swedish National Board of In
dustrial and Technical Development under grant P
Contents
Intro duction and Examples
Intro duction
Geometry of matrix space
Motivation a singular value puzzle
Intro duction to Versal Deformations
Characteristic p olynomials give the feel of versal deformations
The rational canonical form is not enough for derogatory matrices
Versal deformation the linearized theory
Versal deformations the bigger picture
Versal deformations for the Jordan canonical form
The Algebra of Matrix Pencils Canonical Forms
Kronecker canonical form
Generalized Schur form and reducing subspaces
Generic and non generic Kronecker structures
The Geometry of Matrix Pencil Space
The orbit of a matrix p encil and its tangent and normal spaces
A lower b ound on the distance to a less generic p encil
Versal Deformations for the Kronecker Canonical Form
An intro ductory example
Notation a glossary of To eplitz and Hankel matrices
Versal deformations the general case
Applications and Examples
Some examples of versal deformations of matrix p encils in KCF
Versal deformations of the set of by matrix p encils
Using GUPTRI in a random walk in tangent and normal directions
of non generic p encils
Versal deformations and minimal p erturbations for changing a non
generic structure
Conclusions
Acknowledgements
References
Intro duction and Examples
Intro duction
Traditionally canonical structure computations take as their input some mathematical ob
ject a matrix or a p encil say and return an equivalent ob ject that is p erhaps simpler or
makes clear the structure of the equivalence relation Some example equivalence relations
and corresp onding canonical forms are
Structure Equivalence Relation Canonical Form
Square Matrices A X AX Jordan Canonical Form
Rectangular Matrices A U AV Singular Values
Rectangular Matrices A XA Reduced Echelon form
Matrix Pencils A B P A B Q Kronecker Canonical Form
k
Analytic real functions f x f x x
In the rst three examples the input is a matrix in the next example the input is
a p encil In these cases X P and Q are presumed nonsingular and U and V are pre
sumed orthogonal We presume the real functions f are analytic in a neighb orho o d of zero
f and x is monotonic and analytic near zero
Canonical forms app ear in every branch of mathematics A few examples from control
theory may b e found in However researchers in singularity theory have
asked the question what happ ens if you have not one ob ject that you want to put into
a normal form but rather a whole family of ob jects nearby some particular ob ject and
you wish to put each memb er of the family into a canonical form in such a way that the
canonical form dep ends smo othly on the deformation parameters
For example one may have a one parameter matrix deformation of A which is simply
an analytic function A for which A A An n parameter deformation is de ned
n
the same way except that R Similarly one may have n parameter deformations of
p encils or functions Sticking with the matrix example we say two deformations A and
B are equivalent if A and B have the same Jordan canonical form for each and
every A deformation of a matrix is said to b e versal if lo osely sp eaking it captures all
p ossible Jordan form b ehaviors near the matrix A deformation is said to b e miniversal if
it do es so with as few parameters as p ossible A more formal discussion of these de nitions
may b e found in Section
Derivation of versal and miniversal deformations requires a detailed understanding of
the p erturbation theory of the ob jects under study In particular one needs to understand
the tangent space of the equivalence relation and how it is emb edded in the entire space
In Section we explain the mechanics of this p erturbation theory
While we b elieve that versal deformations are interesting mathematical ob jects this
work di ers from other works on the sub ject in that our primary goal is not so much the
versal deformation or the miniversal deformation but rather the p erturbation theory and
how it in uences the computation of the Kronecker canonical form As such we tend to b e
interested more in metrical information than top ological information Therefore we obtain
new distance formulas to the space of less generic matrix p encils in Section In Section
we derive an explicit orthogonal basis for the normal space of a Kronecker canonical form
For us a versal decomp osition will b e an explicit decomp osition of a p erturbation into its
tangential and normal comp onents and we will not derive any miniversal deformations that
may have simpler forms but hide the metric information
Versal deformations for function spaces are discussed in The rst appli
cation of these ideas for the matrix Jordan canonical form is due to Arnold Further
references closely related to Arnold s matrix approach are and The latter refer
ence also includes applications to di erential equations Applications of the matrix
idea towards an understanding of companion matrix eigenvalue calculations may b e found
in The only other work that we are aware of that considers versal deformations of the
Kronecker canonical form is by Berg and Kwatny who have indep endently derived some
of the normal forms considered in this pap er
Our Section contains a thorough explanation of versal deformations from a linear al
gebra p ersp ective Chapter brie y reviews matrix p encils and canonical forms Chapter
derives the geometry of the tangent and normal spaces to the orbits of matrix p encils
Chapter derives the versal deformations while Chapter gives applications and illustra
tions
Geometry of matrix space
Our guiding message is very simple matrices should b e seen in the mind s eye geometrically
as p oints in n dimensional space A p erfect vision of numerical computation would allow
us to picture computations as moving matrices from p oint to p oint or manifold to manifold
Abstractly it hardly matters whether a vector is a column of numb ers or a geometric
p oint in space However without the interplay of these two representations numerical linear
algebra would not b e the same Imagine explaining how Householder re ections transform
vectors without the geometric viewp oint
By contrast in numerical linear algebra we all know that matrices are geometric p oints
in n dimensional space but it is far rarer that we actually think ab out them this way
Most often matrices are thought of as either sparse or dense arrays of numb ers or they
are op erators on vectors
The Eckart Young or Schmidt Mirsky theorem p gives a feel for the geometric
approach The theorem states that the smallest singular value of A is the Frob enius distance
of A to the set of singular matrices One can not help but to see a blob representing the
set of singular matrices This amorphous blob is most often thought of as an undesirable
part of town so unfortunately numerical analysts hardly ever study the set itself
Demmel has help ed to pioneer the development of geometric techniques for the
analysis of ill conditioning of numerical analysis problems Shub and Smale are applying
geometrical approaches towards the solution of p olynomial systems
We b elieve that if only we could b etter understand the geometry of matrix space our
knowledge of numerical algorithms and their failures would also improve A general program
for numerical linear algebra then is to transfer from pure mathematicians the technology to
understand geometrically the high dimensional ob jects that arise in numerical linear algebra
This program may not b e easy to follow A ma jor di culty is that pure mathematicians pay
a price for their b eautiful abstractions they do not always p ossess a deep understanding
of the individual ob jects that we wish to study This makes technology transfer di cult
Even when the understanding exists somewhere it may b e di cult to recognize or may b e
buried under a heavy layer of notation This makes technology transfer time consuming
Finally even after putting in the time for the excavation the knowledge may still b e di cult
to apply towards the understanding or the improving of practical algorithms This makes
technology transfer from pure mathematics frustrating
Nevertheless our goal as researchers is the quest for understanding which we may then
apply In this pap er we follow our program for the understanding of the Jordan and
Kronecker canonical forms of matrices and matrix p encils resp ectively Many of the ideas
to b e found in this pap er have b een b orrowed from the pure mathematics literature with
the goal of simplifying and applying to the needs of numerical linear algebraists
While this is quite a general program for numerical linear algebra this pap er fo cuses
on a particular goal We analyze versal deformations from the numerical linear algebra
viewp oint and then compute normal deformations for the Kronecker canonical form We
consider b oth of these as stepping stones towards the far more di cult goal of truly under
standing and improving up on staircase algorithms for the Jordan or Kronecker canonical
form These are algorithms used in systems and control theory The structures of these
matrices or p encils re ect imp ortant physical prop erties of the systems they mo del such as
controllability
The user cho oses a parameter to measure any uncertainty in the data The existence
of a matrix or p encil with a di erent structure within distance of the input means that
the actual system may have a di erent structure than the approximation supplied as input
These algorithms try to p erturb their input by at most so as to nd a matrix or p encil
with 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 the geometry of matrix space in order to b egin to understand how
we can supply the correct information to the user With this information we b elieve that
we would then b e able to not only correctly provide the least generic solutions but also
understand how singularities hinder this pro cess Bad solutions may then b e re ned so as
to obtain b etter solutions As the next subsection illustrates the geometry directly a ects
the p erturbation theory
Motivation a singular value puzzle
Consider the following four nearly singular matrices
M M M M
Each of these matrices are distance O from the Jordan blo ck J
What is the smaller of the two singular values of each of M M M and M The answer
is
and M M M M
min min min min
A quick way to verify this algebraically is to notice that the larger singular value of each
matrix is approximately so that the smaller is approximately the absolute determinant
of the matrix Another approach that b ounds the smallest singular value is the combination
of the Eckart Young theorem and the observation that these matrices are singular
M M M M M
When in our four matrices b ecome the singular Jordan blo ck J
As varies from each of the four forms in traces out a line in matrix space The
geometric issue that is interesting here is that the line of matrices traced out as varies is
f In Normal Tangent Tangent g to the set of singular matrices Somehow this feels
like the right explanation for why the smaller singular values are f
g
Let us take a closer lo ok at the set of singular matrices The four parameters found in
a matrix M are b est viewed in a transformed co ordinate system
M x y z w x y z w
w z x
y w z
In this co ordinate system the singular matrices fall on the surface describ ed by the equation
w z xy This is a three dimensional surface in four dimensional space The traceless
singular matrices w fall on the cone z xy in three dimensional space
Our matrix J may now b e represented as and the four lines of matrices
mentioned ab ove are
l f x g
l f y g
l f z g
l f w g
The lines l l and l are all traceless i e the matrices on each of these lines may b e
viewed in the three dimensional space of the cone The line l is not only tangent to the
cone but in fact it lies in the cone The line l is tangent to one of the circular cross sections
of the cone
Figure illustrates l as a stick resting near the b ottom of the cone The line l is a
thin line on the cone through the same p oint
The line l is normal to the cone but it is also tangent to the manifold of singular
matrices One way to picture this in three dimensions is to take the three dimensional slice
Figure Cone of traceless singular matrices with stick representing a tangent
of fw z xy g corresp onding to x i e fw z y g This is a hyp erb oloid with
the Jordan blo ck as a saddle p oint The line is the tangent to the parab ola w y which
rests in the plane z Figure illustrates this line with a cylindrical stick whose central
axis is the tangent Lastly the line l is normal to the set of singular matrices
If we move a distance away from a p oint on a surface along a tangent our distance
to the surface remains O This is what the singular value corresp onding to l and l
is telling us Alternatively if we move normal to the surface as in l the singular value
changes more rapidly O
The cone of singular matrices with w is not only a slice of a large dimensional
space but it is also the closure of the set of matrices similar to J which we denote
orbit J in Section The matrices similar to J are singular and traceless In
fact the only matrix that is singular and traceless that is not similar to J is the matrix
which is the vertex of the cone We further explore this case in Section after we have
de ned versal deformations
We conclude that the geometry of the orbit and in particular the directions of the
tangents and normals to the orbit directly in uence the eigenvalue p erturbation theory
Intro duction to Versal Deformations
This intro duction is designed to b e readable for general audiences but we particularly target
the numerical linear algebra community
The ideas here may b e thought of as a numerical analyst s viewp oint on ideas that were
inspired by Arnold s work on versal deformations of matrices Further elab oration up on
Arnold s versal deformations of matrices may b e found in Chapter and and
These ideas t into a larger context of di erential top ology and singularity theory
Bruce and Giblin have written a wonderfully readable intro duction to singularity theory
emphasizing the elementary geometrical viewp oint After reading this intro duction it is
easy to b e lulled into the b elief that one has mastered the sub ject but a whole further more
advanced wealth of information may b e found in Finally what none of these 4
2
0
−2
−4 2 1 2 0 1 0 −1 −1
−2 −2
Figure Manifold of singular matrices The axis of the cylindrical stick is tangent to the
manifold
references do very well is explain clearly that there is still much in this area that mankind
do es not yet fully understand
Singularity theory may b e viewed as a branch of the study of curves and surfaces
but its crowning application is towards the top ological understanding of functions and
their b ehavior under p erturbations Of course numerical analysts are very interested in
p erturbations as well
Characteristic p olynomials give the feel of versal deformations
Let A b e a di erentiable one parameter family of matrices through A A This is
just a curve in matrix space If A has a complicated Jordan canonical form then very
likely the Jordan canonical form of A is a discontinuous function of The Jordan
canonical form you will rememb er can have nasty ones p opping up unexp ectedly on the
sup erdiagonal It is even more desirable if that function can somehow describ e the kinds
of matrices that are near A
Discontinuities are as unpleasant for pure mathematicians as they are for computers
Therefore Arnold asks what kinds functions of are di erentiable or many times
di erentiable or analytic
One function that comes to mind is the characteristic p olynomial p t det A tI
The co e cients of p are clearly di erentiable functions of no matter how complicated a
Jordan canonical form the matrix A might have In numerical linear algebra we never com
pute the characteristic p olynomial b ecause the eigenvalues are often very p o orly determined
by the co e cients of the characteristic p olynomial Mathematically the characteristic p oly
nomial is a nice function of a matrix b ecause its co e cients unlike the eigenvalues of the
matrix are analytic functions of the entries of the matrix
The characteristic p olynomial is a reasonable representation for the Jordan canonical
form under the sp ecial circumstance that every matrix A is non derogatory i e each
matrix has exactly one Jordan blo ck for each distinct eigenvalue By a reasonable represen
tation we mean here that it actually enco des the Jordan canonical form of A Theoretically
if you know the characteristic p olynomial then you know the eigenvalues with appropriate
multiplicities It follows that there is a unique non derogatory Jordan canonical form See
Wilkinson pp or Note p To rep eat there is a one to one corresp ondence
among the n eigenvalues of a non derogatory matrix the characteristic p olynomial of a non
derogatory matrix and the Jordan canonical form of a non derogatory matrix but only the
characteristic p olynomial is a di erentiable function of the p erturbation parameter The
eigenvalues themselves can have rst order p erturbations with the non di erentiable form
n
for example for an n n matrix A with only one Jordan blo ck J This is a well
n
known example
In the language of numerical linear algebra we would say that a non derogatory matrix
A may b e written in companion matrix form KCK in such a way that di erentiable
p erturbations to the matrix A lead to di erentiable p erturbations to the companion matrix
C Here the matrix K is a Krylov matrix See p Equivalently rst order
p erturbations to the matrix A are manifested as rst order p erturbations to the companion
matrix C When A is a companion matrix this gives a rst order p erturbation theory for
the characteristic p olynomials of nearby matrices This p erturbation theory is computed in
Our story would almost stop here if we were only interested in the Jordan form of non
derogatory matrices We use almost b ecause it would b e a shame to stop here without
explaining the ideas geometrically Even if we did not discuss the geometry we have reasons
to continue on since matrix space is enriched with the derogatory matrices and also we wish
to generalize these ideas ab out the Jordan canonical form to cover the more complicated
case of the Kronecker canonical form
The rational canonical form is not enough for derogatory matrices
In the previous subsection we saw that n parameters were su cient to sp ecify the Jordan
canonical form of any matrix in a small neighb orho o d of a non derogatory matrix What
happ ens if the matrix is derogatory One obvious guess turns out to b e wrong The
usual generalization of the companion matrix form for derogatory matrices is the rational
canonical form If A is derogatory it may b e put in rational canonical form This form may
b e thought of as the direct sum of companion matrices C with dimension m m
i
m The characteristic p olynomial of each C divides the characteristic p olynomial of all the
i
k
preceding C j i Can any nearby matrix b e expressed as the direct sum of companion
j
matrices with dimension m m m in a nice di erentiable manner The answer is
k
generally no though go o d enough to sp ecify the Jordan canonical form of a matrix the
rational canonical form fails to b e p owerful enough to sp ecify the Jordan canonical forms
of all matrices in a neighb orho o d The reason is that there are just not enough parameters
in the rational canonical form to cover all the p ossibilities To have enough parameters we
need a versal deformation
Versal deformation the linearized theory
The linearized picture of a versal deformation is easy to understand We therefore explain
this picture b efore plunging into the global p oint of view The general case may b e nonlinear
but the linearized theory is all that really matters For simplicity we assume that we are in
real n dimensional Euclidean space but this assumption is not so imp ortant
n n
We recall the elementary fact that if S and T are subspaces of R such that S T R
then there exist linear pro jections and that map onto S and T resp ectively
S T
Consider a p oint x S We will investigate all p ossible p erturbations y of x but we will
not b e concerned with p erturbations that are within S itself Psychologically we consider
all the vectors in S to somehow b e the same so there will b e no need to distinguish them
n
Let T b e any linear subspace such that S T R i e any vector may b e written as the
sum of an element of T and an element of S not necessarily uniquely Clearly if t t
k
span T then our p erturb ed vector x y may b e written as
k
X
x y x t something in S
i i
i
where the may b e chosen as linear functions of y We see here what will turn out to b e
i
the key idea of a versal deformation every p erturbation vector may b e expressed in terms
of the and vectors that we are considering to all b e equivalent
i
We now formally intro duce the lo cal picture of versal deformations
l
De nition A linear deformation of the point x is a function de ned on R
A x T
where T t t t are arbitrary directions
l
The choice of the word deformation is meant to convey the idea that we are lo oking
at small values of the and these p erturbations are small deformations of the starting
i
p oint x
De nition A linear deformation A of the point x is versal if for al l linear defor
mations B of the point x it is possible to write
B A
where is a linear function from to with and is a linear
m
l
function from into S with S
P
l
n
t is versal if and only if S T R Clearly We now explain why A x
i i
i
A S T and since B may b e arbitrary it is necessary that span ft g S
i
n
R It is also su cient b ecause we then obtain linear pro jections allowing us to write
B x B B The functions and may b e obtained from and
S T S T
De nition A linear deformation A of the point x is universal or miniversal if it is
versal and has the fewest possible parameters needed for a versal deformation
The numb er of parameters in a miniversal deformation is exactly the co dimension of S
Numerical analysts might prefer taking the t to b e an orthogonal basis for S the subspace
i
p erp endicular to S This provides one natural miniversal deformation Arnold do es not
insist on using S any basis for any subspace of dimension n dim S will do provided that
it intersects S at zero only From the top ological p oint of view this is exactly the same
though of course the numerical prop erties may b e quite di erent
Versal deformations the bigger picture
The previous subsection explained the linear or rst order theory of versal deformations
At this p oint the reader might wonder whether this is just a whole lot of jargon to merely
extend a basis for a subspace to the entire space At the risk of delaying the motivation
until now we decided to make sure that the linear theory is well understo o d
n
We are still in a nite dimensional Euclidean space R but S will no longer b e a at
subspace Instead we wish to consider any equivalence relation such that the orbit of x
orbit x fy jy xg is a smo oth submanifold As an example we might de ne x y to
mean kxk ky k in which case the orbits are spheres In this context the word orbit is
quite natural In n dimensional space p oints may b e thought of as n n matrices and
the orbit is the set of matrices with the same Jordan canonical form
One nal example that we must mention b ecause it explains the origins and signi cance
of singularity theory lives in an in nite dimensional space The vector space is the set of
analytic functions f x for which f We can de ne f g if f x and g x
have the same Taylor expansion at x where is a monotonic analytic function with
The orbit of any function is some complicated in nite dimensional manifold but
the co dimension of the manifold happ ens to b e nite
n
Returning to R we can now cast everything into a nonlinear context
De nition A deformation of the point x is any di erentiable function
A
l
satisfying A x
De nition A deformation A of the point x is versal if for al l deformations B
it is possible to write
B A
in an arbitrarily smal l neighborhood of where is a di erentiable function from
to for which
m
l
The go o d news is that the inverse function theorem lets us express this nonlinear notion
in terms of the linear theory
Theorem A deformation A of x is versal if and only if A is a linear deformation
at the point x on the subspace tan orbit x where A is the linearization of A near x i e
only rst derivatives matter and tan denotes the subspace tangent to the orbit at x
The rigorous pro of may b e found in but the intuition should b e clear near the p oint
x only linear deformations matter and the curvature of the orbit b ecomes unimp ortant
only the tangent plane matters In other words y x only if y is in the orbit of x but to
rst order y x if roughly sp eaking y x s where s is a small tangent vector to the
orbit
Versal deformations for the Jordan canonical form
We b egin with deformations of the matrix A J The p erturbation theory and the
normal and tangent spaces were discussed in Section We will use the same co ordinate
system here
Four parameters are su cient to describ e the most general defor
mation of A
A
The equivalence relation is that of similar matrices and it is easy to see by checking the
trace and determinant that for su ciently small values of we have the equivalence
A B
where is de ned by and It is worth
emphasizing that the equivalence relation do es not work if A is derogatory but this do es
not happ en for small parameters
We then see from De nition that the two parameter deformation B is versal In
fact it is miniversal in that one needs the two parameters From the lo cal theory pictured
in Section we saw that the orbit of J is the two dimensional cone and therefore
the tangent and normal spaces are each two dimensional The numb er of parameters in a
miniversal deformation is always the dimension of the normal space
It is a worthwhile exercise to derive the similarity transformation C for which
A C B C
and then linearize this map for small values of to see which directions fall along the
tangent space to the cone and which directions are normal to the cone
Now consider deformations of A I or A Both matrices are derogatory with
eigenvalues and resp ectively The tangent space do es not exist i e it is zero dimen
sional Any p ossible b ehavior may b e found near I or including a one dimensional
space of derogatory matrices The miniversal deformation of I or is the full deformation
requiring four parameters
The general case has b een worked out by Arnold The tangent vectors to the orbit
of a matrix A are those matrices that may b e expressed as XA A X The normal space
is the adjoint of the centralizer i e the set of matrices Z satisfying
H H
A Z ZA
Let A has p distinct eigenvalues i p with p Jordan blo cks each Let q
i i i
denote the sizes of the Jordan blo cks corresp onding to the eigenvalue q q
i i p
i
Then the dimension of the normal space of A is
i
p p p
i
X X X
j q q q q
j i i i i
i j i
Notice that the values of the distint play no role in this formula The dimension of the
i
normal space of A is determined only by the sizes of the Jordan blo cks of A asso ciated
with distinct eigenvalues If the matrix is in Jordan canonical form then the normal space
consists of matrices made up of To eplitz blo cks whose blo ck structure is completely de
termined by the sizes of the Jordan blo cks for di erent eigenvalues The normal space is
the same for all matrices with the same Jordan structure indep endent of the values of the
distinct eigenvalues so one may as well consider only Jordan blo cks corresp onding to a
eigenvalue This form of the normal space for the zero eigenvalues is a sp ecial case in
Theorem
The Algebra of Matrix Pencils Canonical Forms
We saw in Section that to consider versal deformations one needs a nite or in nite
dimensional space and an equivalence relation on this space For the remainder of this
pap er we consider the nite dimensional Euclidean space of matrix p encils endowed with the
Euclidean metric usually denoted the Frob enius metric in this context The equivalence
relation is that of the strict equivalence of p encils
We consider a matrix p encil A B where A and B are arbitrary m n matrices with
real or complex entries The p encil is said to b e regular if m n and det A B is not
identically zero Indeed the zeros of det A B are the generalized eigenvalues of a
regular p encil Otherwise i e if det A B is identically zero or m n A B is called
singular Two m n p encils A B and A B are strictly equivalent if there exist
constant indep endent of invertible matrices P of size m m and Q of size n n such
that
P A B Q A B
Kronecker has shown that any matrix p encil is strictly equivalent to a canonical diagonal
form that describ es the structure elements of A B including generalized eigenvalues
and eigenspaces in full detail e g see This form is a generalization of the Jordan
canonical form JCF to general matrix p encils
Kronecker canonical form
The Kronecker canonical form KCF of A B exhibits the ne structure elements in
cluding elementary divisors Jordan blo cks and minimal indices and is de ned as follows
m n m m n n
Supp ose A B C Then there exist nonsingular P C and Q C such
that
P A B Q S T
where S diag S S and T diag T T are blo ck diagonal S T is
ii ii
bb bb
m n We can partition the columns of P and Q into blo cks corresp onding to the blo cks
i i
of S T P P P where P is m m and Q Q Q where Q is n n
i i i i
b b
T
Each blo ck M S T must b e of one of the following forms J N L or L
i ii ii j j j
j
First we consider
J and N
j j
J is simply a j j Jordan blo ck and is called a nite eigenvalue N is a j j blo ck
j j
corresp onding to an in nite eigenvalue of multiplicity j The J and N blo cks together
j j
constitute the regular structure of the p encil All the S T are regular blo cks if and
ii ii
only if A B is a regular p encil A B denotes the eigenvalues of the regular part of
A B with multiplicities and is called the spectrum of A B
The other two typ es of diagonal blo cks are
T
and L L
j
j
The j j blo ck L is called a singular block of right or column minimal index j It
j
j T T
is has a one dimensional right null space for any The j j blo ck L
j
a singular block of left or row minimal index j and has a one dimensional left null space
for any The left and right singular blo cks together constitute the singular structure of
the p encil and app ear in the KCF if and only if the p encil is singular The regular and
singular structures de ne the Kronecker structure of a singular p encil
m n
We also have a real KCF asso ciated with real matrix p encils If A B R there
m m n n
exist nonsingular P R and Q R where as b efore P A B Q S T is
blo ck diagonal The only di erence with is the Jordan blo cks asso ciated with complex
conjugate pairs of eigenvalues Let i where are real and If is an
eigenvalue of A B then also is an eigenvalue Let J denote a Jordan blo ck of
j
size j j asso ciated with a complex conjugate pair of eigenvalues here illustrated with
the case j
J
The Jordan blo ck J plays the same role in the real Jordan canonical form as diag J
j j
J do es in the complex JCF Notice that each pair of the j columns of the real P and
j
Q asso ciated with a J blo ck form the real and imaginary parts of the generalized
j
principal chains corresp onding to the complex conjugate pair of eigenvalues
Generalized Schur form and reducing subspaces
In most applications it is enough to transfer A B to a generalized Schur form e g to
GUPTRI form
A B
r r
H
A B P A B Q
r eg r eg
A B
l l
where P m m and Q n n are unitary and denotes arbitrary conforming submatrices
Here the square upp er triangular blo ck A B is regular and has the same regular
r eg r eg
structure as A B i e contains all eigenvalues nite and in nite of A B The
rectangular blo cks A B and A B contain the singular structure right and left
r r
l l
minimal indices of the p encil and are blo ck upp er triangular
A B has only right minimal indices in its Kronecker canonical form KCF indeed
r r
the same L blo cks as A B Similarly A B has only left minimal indices in its
j
l l
T
blo cks as A B If A B is singular at least one of A B and KCF the same L
r r
j
A B will b e present in The explicit structure of the diagonal blo cks in staircase
l l
form can b e found in If A B is regular A B and A B are not present in
r r
l l
and the GUPTRI form reduces to the upp er triangular blo ck A B Staircase
r eg r eg
forms that reveal the Jordan structure of the zero and in nite eigenvalues are contained in
A B
r eg r eg
Given A B in GUPTRI form we also know di erent pairs of reducing subspaces
Supp ose the eigenvalues on the diagonal of A B are ordered so that the
r eg r eg
rst k say are in a subset of the sp ectrum and the remainder are outside Let
A B b e m n Then the left and right reducing subspaces corresp onding to are
r r r r
spanned by the leading m k columns of P and leading n k columns of Q resp ectively
r r
When is empty the corresp onding reducing subspaces are called minimal and when
contains the whole sp ectrum the reducing subspaces are called maximal
Several authors have prop osed staircase typ e algorithms for computing a generalized
Schur form e g see They are numerically stable in the sense that
they compute the exact Kronecker structure generalized Schur form or something similar
of a nearby p encil A B k A A B B k is an upp er b ound on the distance to
E
the closest A A B B with the KCF of A B Recently robust software with error
b ounds for computing the GUPTRI form of a singular A B has b een published
Some computational exp eriments that use this software will b e discussed later
Generic and non generic Kronecker structures
Although the KCF lo oks quite complicated in the general case most matrix p encils have a
quite simple Kronecker structure If A B is m n where m n then for almost all A and
B it will have the same KCF dep ending only on m and n This corresp onds to the generic
case when A B has full rank for any complex or real value of Accordingly generic
rectangular p encils have no regular part The generic Kronecker structure for A B with
d n m is
diag L L L L
where bm dc the total numb er of blo cks is d and the numb er of L blo cks is m mo d d
which is when d divides m The same statement holds for d m n
T T
Square p encils are generically regular i e if we replace L L in by L L
det A B if and only if is an eigenvalue The generic singular p encils of size
n by n have the Kronecker structures
T
diag L L j n
j
n j
Only if a singular A B is rank de cient for some may the asso ciated KCF b e more
complicated and p ossibly include a regular part as well as right and left singular blo cks
This situation corresp onds to the non generic case which of course is the real challenge
from a computational p oint of view
The generic and non generic cases can easily b e couched in terms of reducing subspaces
For example generic rectangular p encils have only trivial reducing subspaces and no gen
eralized eigenvalues at all Generic square singular p encils have the same minimal and
maximal reducing subspaces A non generic case corresp onds to that A B lies in a
particular manifold of the matrix p encil space and that the p encil has nontrivial reducing
subspaces Moreover only if it is p erturb ed so as to move continuously within that manifold
do its reducing subspaces and generalized eigenvalues also move continuously and satisfy in
teresting error b ounds These requirements are natural in many control and systems
theoretic problems such as computing controllable subspaces and uncontrollable mo des
The Geometry of Matrix Pencil Space
In this section we derive formulas for the tangent and normal spaces of the orbit of a matrix
p encil that we will make use of in order to compute the versal form in the next section We
also derive new b ounds for the distance to less generic p encils
The orbit of a matrix p encil and its tangent and normal spaces
Any m n matrix pair A B with real or complex entries de nes a manifold of strictly
equivalent matrix p encils in the mn dimensional space P of m by n p encils
orbit A B fP A B Q det P det Q g
We may cho ose a sp ecial element of orbit A B that reveals the KCF of the p encil
As usual the dimension of orbit A B is equal to the dimension of the tangent
space to the orbit at A B here denoted tan A B By considering the deformation
I X A B I Y of A B to rst order term in where is a small scalar
m n
we obtain A B X A B A B Y O from which it is evident that
tan A B consists of the p encils that can b e represented in the form
T T XA AY XB BY
A B
where X is an m m matrix and Y is an n n matrix
Using Kronecker pro ducts we can represent the mn vectors T T tan A B
A B
as
T
I A vec T A I
n A m
vec X vec Y
T
I B B I vec T
n m B
In this notation we may say that the tangent space is the range of the mn m n
matrix
T
A I I A
m n
T
T
B I I B
m n
We may de ne the normal space nor A B as the space p erp endicular to tan A B
Orthogonality in P the mn dimensional space of matrix p encils is de ned with resp ect to
a Frob enius inner pro duct
H H
A B C D tr AC BD
where tr X denotes the trace of a square matrix X Rememb ering that the space orthog
onal to the range of a matrix is the kernel of the Hermitian transp ose we have that
A I B I
m m
H
nor A B ker T ker
H H
I A I B
n n
In ordinary matrix notation this states that Z Z is in the normal space of A B if
A B
and only if
H H H H
Z A Z B and A Z B Z
A B A B
The conditions on Z and Z can easily b e veri ed and also b e derived in terms of the
A B
Frob enius inner pro duct i e
H H H H
T T Z Z tr X AZ BZ Z A Z B Y
A B A B
A B A B
Veri cation if the conditions are satis ed it follows from that the inner pro duct
H H H
is zero Derivation if T T Z Z then tr X AZ BZ Z A
A B A B
A B A
H
Z B Y must hold for any X of size m m and Y of size n n By cho osing
B
H H
B Y which holds for any Y if and only if A Z X reduces to tr Z
B A
H H H H
Z A Z B Similarly we can chose Y which gives that AZ BZ
A B A B
H
If B I this reduces to Z nor A if and only if Z centralizer A which is a
A
A
well known fact e g see We will see in Section that though the A part of the
normal space is very simple when B I obtaining an orthonormal basis for the B part is
H
particularly challenging The requirement that Z A Z when B I destroys any
B A
orthogonality one may have in a basis for the A part
We now collect our general statements and a few obvious consequences
Theorem Let the m n pencil A B be given De ne the mn m n matrix
T as in Then
tan A B range T f XA AY XB BY g
where X and Y are compatible square matrices and
H
nor A B ker T fZ Z g
A B
H H H H
where Z A Z B and A Z B Z
A B A B
The dimensions of these spaces are
dim tan A B m n dim ker T
and
H
dim nor A B dim ker T dim ker T m n
Of course the tangent and normal spaces are complementary and span the complete
mn dimensional space i e P tan A B nor A B so that the dimensions in
and add up to mn as they should
Theorem leads to one approach for computing a basis for nor A B from the
singular value decomp osition SVD of T Indeed the left singular vectors corresp onding
to the zero singular value form such a basis The dimension of the normal space is also
known as the codimension of the orbit here denoted co d A B Accordingly we have
the following compact characterization of the co dimension of orbit A B
Corollary Let the m n pencil A B be given Then
co d A B the numb er of zero singular values of T
The corresp onding result for the square matrix case is
T
co d A the numb er of zero singular values of I A A I
n n
Although the SVD based metho d is simple and has nice numerical prop erties backward
stability it is rather costly in the numb er of op erations Computing the SVD of T is an
O m n op eration
Knowing the Kronecker structure of A B it is also p ossible to compute the co dimen
sion of the orbit as the sum of separate co dimensions
co d A B c c c c c
Jor Jor Sing Sing
Right Left
The di erent contributions in originate from the Jordan structure of all eigenvalues
including any in nite eigenvalue the right singular blo cks L L the left singular
j
k
T T
blo cks L L interactions of the Jordan structure with the singular blo cks L and
k
j
k
T T
and interactions b etween the left and right singular structures L L resp ectively L
j
j
k
Explicit expressions for these co dimensions are derived in Assume that the given A B
has p min m n distinct eigenvalues i p with p Jordan blo cks each Let
i i
denote the sizes of the Jordan blo cks corresp onding to the q q q
i i i p
i
eigenvalue Then the separate co dimensions of can b e expressed as
i
p p p
i
X X X
c j q q q q
Jor j i i i i
j i i
X X X
c j k c j k c j k
Sing
Right Left
j k j k j k
c size of complete regular part numb er of singular blo cks
Jor Sing
Notice that if we do not wish to sp ecify the value of an eigenvalue the co dimension count
i
for this unsp eci ed eigenvalue is one less i e
q q q
i i i
This is sometimes done in algorithms for computing the Kronecker structure of a matrix
p encil where usually only the eigenvalues and are sp eci ed and the remaining ones
are unsp eci ed
It is p ossible to extract the Kronecker structure of A B from a generalized Schur
decomp osition in O max m n op erations The most reliable SVD approach for com
puting a generalized Schur decomp osition of A B requires at most O max m n
op erations which is still small compared to computing the SV D of T for already
mo derate values of m and n e g when m n
For given m and n the generic p encil has co dimension i e span the complete mn
dimensional space while the most non generic matrix pair A B has
m n m n
co dimension mn i e de nes a p oint in mn dimensional space Accordingly any
m n non generic p encil di erent from the zero p encil has a co dimension and mn
A lower b ound on the distance to a less generic p encil
The SVD characterization of the co dimension of orbit A B in Corollary leads to the
following theorem from which we present an interesting sp ecial case as a corollary
Theorem For a given m n pencil A B with codimension c a lower bound on the
distance to the closest pencil A A B B with codimension c d where d is
given by
mn
X
A
p
k A B k T
E
i
m n
i mn c d
where T denotes the ith largest singular value of T T T
i i i
Pro of It follows from Corollary that T has rank mn c if and only if A B
has co dimension c and A A B B has co dimension c d d if and only if
T T where T is de ned as
T
A I I A
m n
T
T
B I I B
m n
p
m nk A B k has rank mn c d From the construction it follows that k T k
E E
each element a and b app ears m n times in T The Eckart Young and Mirsky
ij ij
theorem for nding the closest matrix of a given rank e g see gives that the size of
the smallest p erturbation in Frob enius norm that reduces the rank in T from mn c to
mn c d is
mn c
X
A
T
i
i mn c d
Moreover A B has co dimension c implies that T T Since
mn c mn
k T k must b e larger than or equal to the quantity the pro of is complete 2
E
Corollary For a given generic m n pencil A B a lower bound on the distance to
the closest non generic pencil A A B B is given by
T
min
p
k A B k
E
m n
where T T denotes the smal lest singular value of T which is non zero for a
min mn
generic A B
We remark that the set of m n matrix p encils do es not include orbits of all co dimensions
from to mn
One application of Corollary is to characterize the distance to uncontrollability
for a multiple input multiple output linear system E x t F x t Gu t where E and
F are p by p matrices G is p by q p q and E is assumed to b e nonsingular If
A B GjF E is generic the linear system is controllable i e the dimension
of the controllable subspace equals p and a lower b ound on the distance to the closest
uncontrollable system is given by
Versal Deformations for the Kronecker Canonical Form
In this section we derive versal deformations which for us will mean the decomp osition
of arbitrary p erturbations into the tangent and normal spaces of the orbits of equivalent
p encils
An intro ductory example
We start with a small example b efore considering the general case Let A B L L
with co dimension This means that the manifold orbit A B has co dimension or
dimension in the dimensional space of p encils Since A B already is in KCF
we know its blo ck structure
A B
From the matrices in the tangent space are given by T T XA AX
A B
XB BX where
y x y y x y x y x y x y
y x y y x y x y x y x y
T
y x y y x y x y x y x y
A
x y y x y x y x y x y y
y x y y x y x y x y x y
and
x y y x y x y x y x y y
x y y x y x y x y x y y
x y y x y x y x y x y y
T
B
x y y x y x y x y x y y
x y y x y x y x y x y y
By insp ection we nd the following two relations b etween elements in T and T
A B
a a b b
t t t t
and
a a b b
t t t t
b a
denote the i j th elements of T and T resp ectively These two relations and t where t
A B
ij ij
show clearly that the tangent space has co dimension at least two It may b e veri ed that
the other parameters may b e chosen arbitrarily so that the co dimension is exactly two
We want to nd Z Z that is orthogonal to T T with resp ect to the Frob enius
A B A B
inner pro duct i e
X
a b b a H H
z t z t T T Z Z tr T Z T Z
A B A B A B
ij ij ij ij A B
i j
This inner pro duct is most easily envisioned as the sum of the elementwise multiplication
of the two p encils Using this p oint of view it is obvious that the normal space consists of
p encils of the form Z Z nor A B
A B
p
Z Z
p p p
A B
p p p
p
p
p p p
p p p
p
where p and p are arbitrary Roughly sp eaking the parameter p corresp onds to the
doubly b oxed entries and the parameter p corresp onds to the singly b oxed entries
Now A B Z Z may b e thought of as a versal deformation or normal form with
A B
minimum numb er of parameters equal to the co dimension of the original p encil It follows
that any complex p encil close to the given A B can b e reduced to the parameter
normal form A B Z Z where A B is in Kronecker canonical form
A B
Notation a glossary of To eplitz and Hankel matrices
The example in the previous section shows that a non zero blo ck of Z Z has a struc
A B
tured form Indeed the blo ck has a To eplitz like form with j i non zero
diagonals starting from the element of the blo ck A closer lo ok shows that the
A part has i j non zero diagonals and the B part is just the same matrix negated
and with the diagonals shifted one row downwards In general di erent non zero blo cks
with To eplitz or Hankel prop erties will show up in Z Z nor A B To simplify
A B
the pro of of the general case we intro duce some To eplitz and Hankel matrices Arrows and
stops near the matrices make clear how the matrix is de ned
L
b e a lower trapezoidal s by t To eplitz matrix with the rst non zero diagonal Let S
s t
starting at p osition
p
p
p
L L
S if s t and S otherwise
s t s t
p
s t
p p
s
p p
s s t
L
b e a lower trapezoidal s by t To eplitz matrix with the rst non zero diagonal s and let T
s t
last element at p osition s t
p p
t s
L L
T if s t and T otherwise
s t s t
p
p p p
t t s
p p
t
L
are zero Similarly if s t the entries If s t the entries of the last t s columns of S
s t
L
are zero of the rst s t rows of T
s t
B
b e a banded lower trapezoidal s by t To eplitz with last row Let S
s t
p
p
B B
if s t and S otherwise S
s t s t
p
s t
p
s t
B
b e another banded lower trapezoidal s by t To eplitz matrix this time with last and let T
s t
column
p p
t s
B B
T otherwise if s t and T
s t s t
p p
t s
B B
if s t have all if s t and the last column of T Notice that the last row of S
s t s t
entries equal to zero
L
Moreover let H b e a lower trapezoidal s by t Hankel matrix with the rst non zero
s t
diagonal starting at p osition t
p
p
p
L L
H if s t and H otherwise
s t s t
p
s t
p p
s
p p
s t s
U
b e a similar upper trapezoidal s by t Hankel matrix and let H
s t
p p
t
p p p
t t s
p
U U
H if s t and H otherwise
s t s t
p p
t s
L
are zero Similarly if s t the entries If s t the entries of the rst t s columns of H
s t
U
are zero of the last s t rows of H
s t
Let H b e a dense s by t Hankel matrix with the rst diagonal starting at p osition
s t
p p p p
t
p
H
s t
p
p p
s s t
for b oth the cases s t and s t
The nilpotent k by k matrix
I
k
C
k
will b e used as a shift op erator For a given k by n matrix X the rows are shifted one
T
row upwards and downwards by the op erations C X and C X resp ectively The columns
k
k
are shifted one column rightwards and one column leftwards in an n by k matrix X by the
T
op erations XC and XC resp ectively The k by k matrices
k
k
G I and G I
k k k k
will b e used to pick all rows but one or all columns but one of a given matrix X in the
following way The rst k and last k rows in a k by n matrix X are picked by G X
k
and G X resp ectively The k rst and k last columns in an n by k matrix X are
k
T T
picked by XG and X G resp ectively
k k
Let I denote the k by k matrix obtained by reversing the order of the columns in the
k
k by k identity matrix It follows that for an n by k matrix X the order of the columns is
reversed by the multiplication X I
k
So far the matrices intro duced are rectangular To eplitz and Hankel matrices with a
L L L B B
sp ecial structure e g lower trap ezoidal S T H banded lower trap ezoidal S T
U
upp er trap ezoidal H or dense H The matrices C and G G that will b e used as shift
and pick op erators resp ectively are To eplitz matrices with only one non zero diagonal
In the next section we will see that versal deformations for all combinations of di erent
blo cks in the KCF except Jordan blo cks with non zero nite eigenvalues can b e expressed
in terms of these matrices To cop e with non zero nite Jordan blo cks J we need
k
L L
to intro duce three more matrices First two lower triangular To eplitz matrices D and E
D
which are involved in the case with two J blo cks Finally the monstrous matrix F
k
which captures the cases with a left or right singular blo ck and a J blo ck
k
Given f g de ne two in nite sequences of numb ers d and e by the recursion
i i
d d
i i
e i e
i i
starting with
d
e
Given sizes s and t for q minfs tg we de ne D q and E q as lower
s t s t
triangular To eplitz matrices with q diagonals in terms of d d and e e and a
q q
d boundary value e
q
q
e
d
q
q
d e
q q
and E q D q
s t s t
d e
d d d d e e e e
q q q
q
We take linear combinations with parameters p to form the matrices
j
minfs tg minfs tg
X X
L L
p E i i p D i i and E D
j s t j s t
s t s t
i i
Q
i
k k is de ned to b e and where j minfs tg i and i
k
for i and i resp ectively The parameter index j and the scaling function i are
L L L T L
chosen to satisfy D S and E C S for in Theorem see tables
s t s t s t s s t
and By simplifying using i j and i this consistency will b e lost but we
will still have valid expressions for the versal deformations
L L
The relations b etween the elements of D and E are most readily shown by an
s t s t
example
2 4
4j j 2j j
p
1
3 3
L
4 2 2
D
2
2j j 4j j 2 j j
2
4 3
j j p p p
1 2 1
3 3 3 3
2 2 4
2
2
4j j 2 j j 2j j
2 2
p p p p p j j p
1 2 3 1 2 1
3 3 3 3 3
and
4 2
4 j j j j 2
p
1
3 3
L
4 2 2
E
2
2 j j j j 2j j 4
4 3
j j p p p
1 1 2
3 3 3
2 4 2
2
2j j 2 4 j j j j
2
p p j j p p p p
1 1 1 2 3 2
3 3 3 3
D
D for dense b e de ned as Let F
s t
s
X
D
F p F i
s i s t
s t
i
where F q has the q last rows non zero and de ned as
s t
j
f for j t
s q j
f f for i s q s j t f
i j i j i j
and f for i s q s is de ned as the solution to
i
T T T T
F q G F q G F s i G F s i G
s t s t s t s t
t t t t
Notice that f is used as an unknown in the generation of elements in In the
i
T
de nition of F q the solutions for f for i s q s ensure that F q G
s t i s t
t
T T T
for q q F q G is orthogonal to F q G F q G
s t s t s t
t t t
Also here we show a small example to facilitate the interpretation of the de nition
p p
1 1
2 2
(j j +1) j j +2
D
p p p p
F
4 2 4 2
2 1 2 1
3 2
j j +2j j +2 j j +2j j +2
2
1
p p p p p p
2 4 2 2 4 2
2 3 2 1 3 1
j j +1 j j +2j j +2 j j +1 j j +2j j +2
Versal deformations the general case
Without loss of generality assume that A B already is in Kronecker canonical form
M diag M M M where each M is either a Jordan blo ck asso ciated with a nite
b k
or in nite eigenvalue or a singular blo ck corresp onding to a left or right minimal index A
p encil T T XM MY in the tangent space can b e partitioned conformally with
A B
B A
X M M Y where M is m by n X is m by m T the p encil M so that T
ij j i ij ij i j
k k k
ij ij
and Y is n by n
ij i j
X X Y Y M M
b b
M M X X Y Y
b b b bb b bb
A B
Since the blo cks T T i j b are mutually indep endent we can study
ij ij
B A
b e conformally sized blo cks Z the di erent blo cks of T T separately Let Z
A B
ij ij
of Z Z From we know that Z Z is in the normal space if and only if
A B A B
H H H H
A Z B Z and Z A Z B We obtain a simple result since A and B are
A B A B
blo ck diagonal
Prop osition Assume that M A B diag A A A diag B B B
b b
is in Kronecker canonical form where each block A B M represents one block in the
i i i
Kronecker structure Then Z Z nor A B if and only if
A B
H A H B A H B H
A Z B Z and Z A Z B for i b and j b
j j i j j i j i i j i i
The mutual indep endency of the i j blo cks of Z and Z implies that we only have
A B
to consider two M blo cks at a time
k
B B A A
T T T T
Y Y M M X X
ii ij i i ii ij
ij ii ij ii
T i j T i j
A B
B B A A
T T T T Y Y M M X X
j i j j j j j i j j
j j j i j j j i
and
B B A A
Z Z Z Z
ij ii ij ii
Z i j Z i j
A B
B B A A
Z Z Z Z
j j j i j j j i
Notably by interchanging the blo cks M A B and M A B in the KCF we
i i i j j j
only have to interchange the corresp onding blo cks in Z Z accordingly For example
A B
if Z i j Z i j in b elongs to nor diag M M then
A B i j
B B A A
Z Z Z Z
j i j j j i j j
nor diag M M
j i
B B A A
Z Z Z Z
ii ij ii ij
This implies that given two blo cks M and M it is enough to consider the case
i j
diag M M In the following we will order the blo cks in the KCF so that Z Z
i j A B
is blo ck lower triangular
Theorem Let A B diag A A A diag B B B be in KCF with
b b
the structure blocks M A B ordered as fol lows L J J for f g
i i i
k k k
T
N and L where the ordering within each block type is in increasing order of size except
k
k
T
for the L blocks which are ordered by decreasing order of size
k
For al l i and j let the i j j i and i i j j blocks of Z Z corresponding to
A B
diag M M be built from Table and Table respectively
i j
Then Z Z gives an orthogonal basis for nor A B with minimum number of
A B
parameters
Table Blo cks in Z Z nor A B where it for L L J J J
A B
T T
J and N N is assumed that For L L is assumed Also
is assumed
A B A B
KCF M M Z Z Z Z
i j
ij ij j i j i
B T B
L L S C S
L T L
L J S C S
D T D T
L J F G F G
T L L
L N C H H
T
G H L L G H
L L L T L T
S T J J S C T C
T U U T
C J L H H
L L L L
J J D E D E
T D T D T
J L G I F G I F
T L T L L L
N N C S C T S T
T L T L
N L T C T
T T B B
L L T T C
J J
J N
J J
J N
Table The diagonal blo cks in Z Z nor A B
A B
A B
KCF M Z Z
i
ii ii
L
L L T
S J S C
L L
J D E
L T L
S N C S
T
L
The sup erscripts B L U and D of the matrices in tables and are parts of the matrix
de nitions in Section The sup erscript T is matrix transp ose All subscripts e g
refer to the sizes of the matrices
Notice that the diagonal blo cks i i and j j of Z Z can also b e obtained from
A B
Table by setting i j For clarity we also display the expressions for the i i and j j
blo cks of Z Z corresp onding to all kinds of structure blo cks M in Table Of course
A B i
the j j blo cks corresp onding to M are read from Table by substituting with
j
The pro of of Theorem consists of three parts
The blo cks of Z Z displayed in Table ful ll the conditions in Prop osition
A B
which imply that Z Z nor A B is orthogonal to an arbitrary T T
A B A B
tan A B
The numb er of indep endent parameters in Z Z is equal to the co dimension
A B
of orbit A B which implies that the parameterized normal form has minimum
numb er of parameters
Each blo ck in Table de nes an orthogonal basis i e the basis for each parameter
p is orthogonal to the basis for each other parameter p i j
i j
We start by proving part followed by proving parts and for the di erent cases
diag M M corresp onding to di erent combinations of structure blo cks in the KCF In
i j
Table we display the co dimension for these cases and the numb er of parameters in the
i i i j j i and j j blo cks of Z Z The co dimensions are computed from
A B
which is the minimum numb er of parameters required to span the corresp onding normal
space For the ordering and the sizes of the blo cks in A B we have made the same
assumptions in Table as in Table Notice that the co dimension counts for L L and
T T
L L are if The numb er of parameters required in each of the i i i j j i
and j j blo cks of Z Z follows from the pro of given b elow
A B
Pro of of part We show that each matrix p encil blo ck in Table has all its param
eters in orthogonal directions This is trivial for blo cks built from the structured To eplitz
L B L U L B
and Hankel matrices S S H H H T or T p ossibly involving some kind of
shift Rememb er that the Frob enius inner pro duct can b e expressed in terms of the sum of
all results from elementwise multiplications as shown in For each of these matrices
the elementwise multiplication of the basis for one parameter p and the basis for another
i
parameter p j i only results in multiplications where at least one of the two elements
j
is zero Obviously these bases are orthogonal For the matrix p encil blo cks built from
D
the F matrix the orthogonality follows from construction since some of the elements are
explicitly chosen so that the Frob enius inner pro duct is zero
L L
For the pro of for the blo cks of typ e D E we de ne s in terms of the d and e in
q i i
to b e
q q
X X
d e s ijd j ije j q
q q q i i
i i
Indep endent of s and t the numb er s is the inner pro duct of the q th basis vector with the
q
r th where q r
We show by induction that s for q Clearly s j j
q
Table The numb er of parameters in the i i i j j i and j j blo cks of Z Z
A B
nor M M
i j
KCF M M co d M M i i i j j i j j
i j i j
L L
L J
L J
L N
T
L L
J J
T
J L
J J
T
J L
N N
T
N L
T T
L L
J J
J N
J J
J N
We now show that s s from which the result follows
q q
d e q jd j q je j q d e q
q q q q q q
e d e q d d e q
q q q q q q
d q d e q e
q q q q
e
q
d e d e e q d q
q q q q q q
q
d q e e q e
q q q q
Since Z Z is built from b mutually indep endent blo cks in Table each asso ciated
A B
with c parameters it follows that Z Z is an orthogonal basis for a c c c
2
i A B
b
dimensional space with one parameter for each dimension 2
Pro of of parts and Now it remains to show that Z Z is orthogonal to
A B
tan A B and that the numb er of parameters in Z Z is equal to co d A B
A B
Since the numb er of parameters in orthogonal directions cannot exceed the co dimension it
is su cient to show that we have found them all The orthogonality b etween Z Z and
A B
A H
Z tan A B is shown by proving that each pair of blo cks ful lls the conditions A
j i j
H B H A B H
in Prop osition In the following we refer to these as B Z A and Z Z B
i j i i j i j i j
the rst and second conditions resp ectively
We carry out the pro ofs for all cases M M in Table starting with blo cks where
i j
M and M are of the same kind
i j
J J We note that J C I First condition for the j i blo ck
k k k
H A T L T L H B
A Z C T I C T B Z
j j i j j i
Second condition for the j i blo ck
A H L T L T T L B H
Z A T C T C I C T I Z B
j i i j i i
L T T L
where we used that T C C T for Similarly for the i j blo ck
H A T L T L H B
A Z C S I C S B Z
i ij i ij
and
A H L T L T T L B H
Z A S C S C I C S I Z B
ij j ij j
L L T T
S for Here we used that S C C
Since the i i i j and j i blo cks of Z Z have parameters each and the
A B
j j blo ck has parameters the total numb er of parameters in Z Z is equal to
A B
co d J J
N N Since there is a symmetry b etween J C I and N I C and
k k k k k k
there is a corresp onding symmetry b etween blo cks in Z Z for J and N blo cks
A B
k k
the pro of for N N is similar to the case J J
J J Here the j i blo ck and the i j blo ck are de ned similarly see Table
and therefore it is su cient to prove one of them with no constraints on and We
note that J I C I We show that the rst and second conditions hold for
k k k k
B A
E q for q minf g First condition D q and Z Z
j i j i
T H A H
D q C D q A Z I C D q
j j i
Rememb er that D q has all elements zero except for the q lower left diagonals where
all elements in each diagonal are identical and de ned by the element in the rst column
A H
gives the following matrix All diagonals Z For q the pro of is trivial For q A
j i j
starting at p osition u for u q are zero The elements in the diagonal
d which by de nition is equal to e which in turn starting at p osition q are
q
q
de nes the corresp onding diagonal in E q The elements in the diagonals starting at
d d Since d is de ned as p ositions u where u q are equal to
u u u
d e the elements in these diagonals are equal to e which de nes the elements in
u u u
B H
we have proved the Z the corresp onding diagonals in E q Since E q B
j i j
rst condition
Second condition Since D q only has q minfs tg non zero diagonals in the lower left
corner of the matrix a shift of rows downwards gives the same result as a shift of columns
T T
Using information from the rst part we obtain leftwards i e C D q D q C
T T A H H
D q D q D q C D q C Z A D q I C
j i i
H A B H
Z E q Z B A
j j i j i i
since B is the identity matrix
i
B A
is and it is parameters in each Z Also here the numb er of parameters in Z
j j j j
of the other three blo cks giving in total
Even though the i i j i i j and j j blo cks lo ok rather complicated they reduce
for to the corresp onding blo cks for J J in Table
L L Here we use L G G First condition for the j i blo ck
k k k
T B
C S
H A T B T T B H B
A Z G S G C S B Z
B
j j i j j i
S
Second condition for the j i blo ck
T B
C S
A H B T T B T B H
Z A S G C S G Z B
B
j i i j i i
S
Since the contribution from L L to the co dimension is and the j i blo ck has
indep endent parameters we deduce that all other blo cks in Z Z are zero
A B
T T
L L Since this case is just the transp ose of L L the pro of is almost the same
and therefore we omit the technical details here
So far we have proved all cases where b oth blo cks are of the same typ e Since the
diagonal blo cks in Z Z always corresp ond to such cases see Table for the numb er
A B
of parameters in these blo cks we from now on only have to consider the i j and j i
blo cks where i j for the remaining cases
L J First condition for the j i blo ck
H A T L T L H B
A Z C S I C S B Z
j j i j j i
Second condition for the j i blo ck
T B H T T L A H L
G Z B G C S Z A S
j i i j i i
The i i and j j blo cks contribute with zero and parameters resp ectively Since
the j i blo ck gives another parameters we have found all parameters and therefore
B A
Z it follows that Z
ij ij
L J First condition for the j i blo ck
D T T D T H A H D T
F C F G G A Z I C F G
j j i
d d
if u and f f By insp ection we see that the u v element of this matrix is
u v
u v
d d D
f if u where f denotes the u v element of F The right hand side of the
u v u v
same condition is
H B D T
B Z I F G
j j i
D
which simply is the leftmost columns of F The u v element of this matrix is
d d d d
which is de ned as then f if u and f if u f f
u v u v
u v u v
Second condition for the j i blo ck
A H T D T T D D T B H
Z A G F G I G F F G G Z B
j i i j i i
As in the previous case the i i and j j blo cks contribute with zero and parameters
resp ectively Since the j i blo ck gives the remaining parameters the i j blo ck is the
zero p encil
Notably for the monstrous j i blo ck reduces to the j i blo ck for L J
in Table
L N First condition for the j i blo ck
H A T L T L H B
A Z I C H C H B Z
j j i j j i
Second condition for the j i blo ck
A H T T L L T B H
Z A G C H H G Z B
L
j i i j i i
H
Also here the i i and j j blo cks contribute with zero and parameters resp ectively
Since the j i blo ck gives the remaining parameters the i j blo ck is the zero p encil
T
L L For this case the i i and j j blo cks are zero p encils First condition for the
j i blo ck
H A H B
A Z G G H I H G G H B Z
j j i j j i
Second condition for the j i blo ck
A H T
Z A G H G
j i i
which is a matrix consisting of the rst rows and last columns of H
This matrix is identical to the one given by the last rows and rst columns of
H i e
T B H
G H G Z B
j i i
Since this blo ck has all parameters it follows that the i j blo ck is the zero
p encil
T
J L First condition for the j i blo ck
H A U
Z G H A
j j i
U
which simply is the last rows in H Another way to construct this matrix is to
U
shift the columns in H one column leftwards and pick the rst columns of that
matrix which can b e written as
U T H B
G H C B Z
j j i
Second condition for the j i blo ck
A H U T U T B H
Z A H C H C I Z B
j i i j i i
The i i and j j blo cks contribute with and zero parameters resp ectively Since
the j i blo ck gives another parameters we conclude that the i j blo ck is the zero
p encil
T
J L Since the pro of for this case is similar to the one for the case L J we
omit the technical details here It follows that for the j i blo ck reduces to the j i
T
blo ck for J L in Table
T
N L First condition for the j i blo ck
H A L T
A Z G T C
j j i
L
shifted one column leftwards This matrix is identical which is the last rows in T
L
to the one given by the rst rows in T which is
L H B
G T B Z
j j i
Second condition for the j i blo ck
A H L T L T B H
Z A T C I T C Z B
j i i j i i
The i i and j j blo cks in Z Z contribute with and zero parameters resp ec
A B
tively Since the j i blo ck gives another parameters we conclude that the i j blo ck
is the zero p encil
J J J N J J and J N In these four cases the i i
and j j blo cks contribute with and parameters resp ectively and therefore the j i
and i j blo cks are zero p encils
Since we have considered all p ossible cases of M and M blo cks the pro of is complete 2
i j
Applications and Examples
Some examples of versal deformations of matrix p encils in KCF
In the following we show three examples of versal deformations of matrix p encils For the
p encil A B L J J with co dimension the parameter versal
deformation A B Z Z where Z Z nor A B is given by
A B A B
p p p
p p p p p p Z
A
p p
p p p p p
p p p p p p p p
and
p p Z p
B
p p
p p p p p
For the p encil A B L J with co dimension the parameter versal
deformation A B Z Z where Z Z nor A B is given by
A B A B
p p p j j
Z
A
2
j j
p p p p p p p j j
2 2
j j j j
and
p p p j j
Z
B
2
j j
p p p p p p j j p
2 2
j j j j
T
with co dimension the For the p encil A B L J N L
parameter versal deformation A B Z Z where Z Z nor A B is given
A B A B
by
p p
p p p p
p p p p p
Z
p p
A
p p p p
p p p p p
p p p p p p
p p p p p p
p p p p p p
and
p p
p p p p
p p
Z
p p p p
B
p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
Versal deformations of the set of by matrix p encils
In the algebraic and geometric characteristics of the set of by matrix p encils were
examined in full detail including the complete closure hierarchy There all non zero and
nite eigenvalues were considered as unsp eci ed R was used to denote a by blo ck with
non zero nite eigenvalues i e any of the three structures J J J J and
J where f g However in the context of versal deformations all these forms
are considered separately and with the eigenvalues sp eci ed known Consequently we now
have di erent Kronecker structures to investigate For example the versal deformation
of A B L J f g is found by computing Z Z
A B
2 2
p p j j p j j p
1 3 3 1
2 2
p p
1 1
p j j p j j p p p p p p
2 2
3 3 2 2 3 4 3 4
j j +1 j j +1
In Table we show the versal deformations for all di erent Kronecker structures for this set
of matrix p encils The di erent structures are displayed in increasing co dimension order
Using GUPTRI in a random walk in tangent and normal directions of
non generic p encils
In order to illustrate how p erturbations in the tangent space and in the normal space a ect
the Kronecker structure computed by a staircase algorithm we have p erformed a set of tests
on non generic by matrix p encils Since the staircase algorithm considers all non zero
nite eigenvalues as unsp eci ed we have not included these cases in the test
For the remaining non generic cases a random p erturbation E E with entries
A B
a b
e e has b een decomp osed into two parts T T tan A B and Z Z
A B A B
ij ij
nor A B such that
E T Z and E T Z
A A A B B B
We illustrate the decomp osition of E E with A B L J From Table we
A B
get
p p
Z Z
B A
p p p p p
Table Versal deformations A B Z Z of the set of by matrix p encils
A B
KCF A B Z Z
A B
L
2
L J
1 1
2
p p p p p p
1 1 1 2 2 1
L J
1 1
p p
1 2
L N
1 1
p p
1 2
p p p p
1 3 3 1 1
1 1
L J J
0 1 1 1 2
p p p p
2 2 4 4 2
2 2
L J See
0 2
p p p p p p
1 3 3 5 5 1
L J
0 1
p p p p p p
2 4 4 6 6 2
p p
1 3
L J J
0 1 1
p p p p
2 2 4 4
p p p p
1 3 3 1
L J N
0 1 1
p p
2 4
p p
1 3
L J
0 2
p p p p p
2 1 4 3 3
p p
1 3
L N
0 2
p p p p p
1 2 3 4 3
p p
1 3
L J N
0 1 1
p p
2 4
T
L L L
0 1
0
p p p p p p
1 2 3 4 4 5
p p p
1 3 5
L J
0 1
p p p
2 4 6
p p p
1 3 5
L N
0 1
p p p
2 4 6
p p p p
1 2 4 5
T
L L
0
1
p p p p
2 3 5 6
p p p p p p
1 4 4 7 7 1
T
L J L
0 1
0
p p p p p p
2 3 5 6 8 8
p p p
1 4 7
T
L J L
0 1
0
p p p p p
2 3 5 6 8
p p p
1 4 7
T
L N L
0 1
0
p p p p p
2 3 5 6 8
p p p p p p
1 2 5 6 9 10
T
L L
0
0
p p p p p p
3 4 7 8 11 12
Let T T E E Z Z Now the parameters p are determined by
A B A B A B i
computing the comp onent of E E in each of the four orthogonal but not orthonormal
A B
directions that span the normal space
Z
Z
Z
Z
We conclude that
b a a b a
e e e e e
a a
p e p p e p
It is easily veri ed that T T Z Z
A B A B
GUPTRI has b een used to compute the Kronecker structure of the p erturb ed
p encils A B E E A B Z Z and A B T T for
A B A B A B
We investigate how far we can move in the tangent and normal
directions b efore GUPTRI rep orts the generic Kronecker structure
The pro cedure has b een rep eated for all cases and for random p erturbations E E
A B
where k E E k and kE k kE k The entries of E E are uniformly dis
A B F A F B F A B
tributed in For each case and for each p erturbation E E we record the
A B
size of when GUPTRI rep orts the generic Kronecker structure In Table we display the
smallest median and maximum values of for the random p erturbations
Entries marked in Table represent that the generic structure was not found for any
size of the p erturbations All these results were for p erturbations in tan A B and they
indicate that for these Kronecker structures there are no or only small curvatures in the
orbit at this p oint p encil Here the tangent directions are very close to orbit A B
Notably the results for the p erturbations E E are except for one case similar
A B
to the results for Z Z This is natural since the p erturbation E E implies a
A B A B
translation b oth in the tangent space as well as the normal space directions The structure
changes app ear more rapidly in the normal space i e for smaller Our computational
results extend the cone example in Section to by matrix p encils
Why is the smallest p erturbation Z Z enough to nd the generic structure
A B
T
The explanation is connected to for the three cases L J L N and L L
the pro cedure for determining the numerical rank of matrices
GUPTRI has two input parameters EPSU and GAP which are used to make rank decisions
in order to determine the Kronecker structure of an input p encil A B Inside GUPTRI
the absolute tolerances EPSUA kAk EPSU and EPSUB kB k EPSU are used in all rank
E E
decisions where the matrices A and B resp ectively are involved Supp ose the singular
values of A are computed in increasing order i e
k k
Table How far we can move in tangent and normal directions b efore non generic by
matrix p encils turn generic
Z Z T T
A B A B
A B co d A B
min max min max
median median
L J
L N
L J
L N
L J N
T
L L L
L J
L N
T
L L
T
L J L
T
L N L
T
L L
then all singular values EPSUA are interpreted as zeros The rank decision is made
k
more robust in practice if EPSUA but EPSUA GUPTRI insists on a gap b etween
k k
the two singular values such that GAP If GAP is also treated
k k k k k
as zero This pro cess is rep eated until an appreciable gap b etween the zero and non zero
singular values is obtained In all of our tests we have used EPSU and GAP
T
b oth the A part and the B part are zero For the most non generic case L L
matrices giving EPSUA EPSUB which in turn lead to the decision that a full rank
p erturbation E E times a very small is interpreted as a generic p encil For the other
A B
two cases either the A part or the B part is full rank and the other part is a zero matrix
which accordingly is interpreted to have full rank already for the smallest p erturbation
Versal deformations and minimal p erturbations for changing a non generic
structure
In the following we illustrate how versal deformations are useful in the understanding of
the relations b etween the di erent structures by lo oking at requirements on p erturbations
to A B for changing the Kronecker structure Assume that we have the following matrix
p encil with the Kronecker structure L J
A B and Z Z
A B
p p
It was shown in that L J with co dimension is in the closure of orbit L J
f g but otherwise unsp eci ed with co dimension which in turn is in the closure
of orbit L the generic KCF with co dimension Notice in Table since is assumed
sp eci ed L J has two parameters and co dimension In the discussion that
follows we assume that is nite non zero but unsp eci ed
We will now for this example illustrate how p erturbations in the normal space directions
can b e used to nd more generic Kronecker structures going upwards in the Kronecker
structure hierarchy and how we can p erturb the elements in A B to nd less generic
matrix p encils Since the space spanned by Z Z is the normal space we must always
A B
rst hit a more generic p encil when we move in nitesimally in normal space directions
The KCF remains unchanged as long as p p but for p and p the
KCF is changed into L J with p That is by adding a comp onent in a normal
space direction we nd a more generic p encil in the closure hierarchy Notably the size of
the required p erturbation is equal to the smallest size of an eigenvalue to b e interpreted as
non zero By cho osing p non zero and p arbitrary the resulting p encil will b e generic
with the KCF L
To nd a less generic structure we may pro ceed in one of the following ways
Find a less generic structure in the closure of orbit L J
Go upwards in the closure hierarchy to a more generic structure and then lo ok in
that orbit s closure for a less generic structure
We know from the investigation in that all structures with higher co dimension than
A B L J include an L blo ck in their Kronecker structures which in turn imply
that A and B must have a common column nullspace of at least dimension Therefore
the smallest p erturbation that turns L J less generic is the smallest p erturbation
that reduces the rank of
A
B
The size of the smallest rank reducing p erturbation is equal to the smallest of the singular
values and By just deleting one the corresp onding p erturb ed p encil is a less
i
generic p encil within the closure of orbit L J These three cases corresp ond to
approach ab ove We summarize these p erturbations and the p erturbations in the normal
space in Table Notice that approach will always require a p erturbation larger than
minf g
i
Which of the non generic structures displayed in Table is obtained by the smallest
p erturbation to L J Mathematically it is easy to see that the p erturbations in the
normal space always can b e made smaller than a rank reducing p erturbation since p
i
and p are parameters that can b e chosen arbitrary small e g smaller than minf g
i
However in nite precision arithmetic it is not clear that the smallest p erturbation
required to nd another structure is in the normal direction This can b e illustrated by
using GUPTRI to compute the Kronecker structures for A B as in and p erturb ed as
in Table For EPSU and GUPTRI uses di erent tolerances
EPSUA and EPSUB for making rank decisions in A and B resp ectively It
follows that for p and p of order GUPTRI still computes the Kronecker structure
L J However if p p and the B part of the p encil is p erturb ed by or
GUPTRI computes the less generic structures just as shown in Table
Table Perturbing A B de ned in yields the p encil A B with more or less
generic structures The co dimension of the original orbit is
k A B k A B KCF co d A B
F
p L
p
L J p p
p
L J N
T
L L L
L J
Conclusions
In this pap er we have obtained not only versal deformations for deformations of Kronecker
canonical forms but more imp ortantly for our purp oses metrical information for the p er
turbation theory of matrix p encils relevant to the Kronecker canonical form In Part II
of this pap er we will explore the strati cation theory of matrix p encils with the goal of
making algorithmic use of the lattice of orbits under the closure relationship
Acknowledgements
We would like to thank Jim Demmel for conveying the message that the geometry of matrix
and matrix p encil space in uences p erturbation theory and numerical algorithms We
further thank him for many helpful discussions In addition the rst author would like to
thank the second two authors for their kind hospitality and supp ort at Ume a University
where this work and fruitful collab oration b egan
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