Computation of the Jordan Normal Form of a Matrix

COMPUTATION OF THE JORDAN NORMAL FORM OF A MATRIX

USING VERSAL DEFORMATIONS

ALEXEI A. MAILYBAEV

Abstract. Numerical problem of nding multiple eigenvalues of real and complex nonsymmet-

ric matrices is considered. Using versal deformation theory, explicit formulae for approximations

of matrices having nonderogatory multiple eigenvalues in the vicinity of a given matrix A are de-

rived. These formulae provide lo cal information on the geometry of a set of matrices having a given

nonderogatory Jordan structure, distance to the nearest matrix with a multiple eigenvalue of given

multiplicity,values of multiple eigenvalues, and corresp onding Jordan chains (generalized eigenvec-

tors). The formulae use only eigenvalues and eigenvectors of the initial matrix A,whichtypically has

only simple eigenvalues. Both the case of matrix families (matrices smo othly dep endentonseveral

parameters) and the case of real or complex matrix spaces are studied. Several examples showing

simplicity and eciency of the suggested metho d are given.

Key words. Jordan normal form, multiple eigenvalue, generalized eigenvector, matrix family,

versal deformation, bifurcation diagram

AMS sub ject classi cations. 65F15, 15A18, 15A21

1. Intro duction. Computation of the Jordan normal form (JNF) of a square

nonsymmetric matrix A is needed in many branches of mathematics, mechanics, and

physics. Due to its great imp ortance this problem was intensively studied during

the last century. It is well known that a generic (typical) matrix has only simple

eigenvalues, which means that its JNF is a diagonal matrix [1,2]. The problem

of nding the JNF of a generic matrix has b een well studied b oth from theoretical

and practical (numerical) sides; see [24]. Nevertheless, manyinteresting and imp or-

tant phenomena, asso ciated with qualitativechanges in the dynamics of mechanical

systems [17,18, 22], stability optimization [3, 14, 20], and eigenvalue b ehavior under

matrix p erturbations [4, 21, 23], are connected with degenerate matrices, i.e., matrices

having multiple eigenvalues.

In the presence of multiple eigenvalues the numerical problem of calculation of the

JNF b ecomes very complicated due to its instability, since arbitrarily small p ertur-

bations (caused, for example, by round-o errors) destroy the nongeneric structure.

This means that we should consider a matrix given with some accuracy,thus, substi-

tuting the matrix A by its neighb or in the matrix space. Suchformulation leads to

several imp ortant problems. The rst problem left op en by Wilkinson [25, 26] is to

nd a distance from a given matrix A to the nearest nondiagonalizable (degenerate)

matrix A. This problem can b e reformulated in the extended form:

to nd a distance (with resp ect to a given norm) from a matrix A to the

nearest matrix A having a given Jordan structure;

to nd a matrix A with the least generic Jordan structure in a given neigh-

b orho o d of the matrix A (the least generic Jordan structure means that a set

of matrices with this Jordan structure has the highest co dimension).

Here we assume that two matrices have the same Jordan structure, if their JNFs di er

only byvalues of di erent eigenvalues. Co dimensions of sets of matrices having the

same Jordan structure were studied by Arnold [1, 2]. According to Arnold, matrices

A with the least generic Jordan structure have the most p owerful in uence on the

Institute of Mechanics, Moscow State Lomonosov University,Michurinsky pr. 1, 117192 Moscow,

Russia ([email protected]). 1

2 ALEXEI A. MAILYBAEV

eigenvalue b ehavior in the neighb orho o d of the matrix A. Therefore, the ab ove stated

problems represent the key p oint in the numerical analysis of multiple eigenvalues.

Several authors addressed the problem of nding b ounds on the distance to the

nearest degenerate matrix; see [5,6,9,15,25,26] and more references therein. A nu-

merical metho d for nding the JNF of a degenerate matrix given with some accuracy

was prop osed in [12,13]. This metho d is based on making a numb er of successive

transformations of the matrix and deleting small elements whenever appropriate. Nev-

ertheless, this pro cedure do esn't provide the least generic JNF, and it is restricted to

the use of the Frob enius matrix norm (for example, it can not b e used in the case,

when di erententries of a matrix are given with di erent accuracy). Another metho d

based on singular value decomp ositions was given in [19]. This metho d is simpler,

but it has the same prop erties. More references of JNF computation metho ds can b e

found in [8].

It is understo o d now that the problem of calculation of the nondiagonal JNF

needs the singularity theory approach based on the use of versal deformations (lo cally

generic families of matrices). Versal deformations were intro duced by Arnold [1, 2] in

order to stabilize the JNF transformation, but no numerical metho ds were suggested.

Fairgrieve[9] succeeded in using the idea of versality for construction of a stable

numerical pro cedure, which calculates matrices of a given Jordan structure. Though

this metho d do es not provide the nearest matrix of a given Jordan structure, it shows

the usefulness of the singularity theory for the numerical eigenvalue problem. In [7]

versal deformations were used for improvement of an existing numerical program in a

similar problem of nding the Kronecker normal form of a matrix p encil. Application

of the singularity theory to computation of a distance from a matrix A to the nearest

matrix with a double eigenvalue [15] showed that this numerical problem is very

complex even in the case of a double eigenvalue.

This pap er is intended to prove that the versal deformation theory can b e an

indep endentpowerful to ol for analysis of multiple eigenvalues. It is shown that us-

ing versal deformations wecanavoid a numb er of transformations and singular value

decomp ositions (used in most algorithms). Instead of this, explicit approximate for-

mulae describing the geometry of a set of matrices with given Jordan structure in

the vicinityofa given matrix A are derived. These formulae use only information

on eigenvalues and eigenvectors of the matrix A. Since the matrix A is typically

nondegenerate, i.e., has only simple eigenvalues, its eigenvalues and eigenvectors can

b e obtained by means of standart co des. As a result, we obtain approximations of a

distance from A to the nearest matrix having a given Jordan structure, its multiple

eigenvalues, and the Jordan chains (generalized eigenvectors). The metho d do es not

dep end on the norm used in the matrix space. Moreover, it allows analyzing multiple

eigenvalues of matrices dep endent on several parameters (matrix families), where we

can not vary matrix elements indep endently.Approximations derived in the pap er

have the accuracy O (" ), where " is the distance from A to a matrix we are lo ok-

ing for. All these prop erties make the suggested approach useful and attractive for

applications.

In this pap er the case of a nonderogatory JNF (every eigenvalue has one corre-

sp onding Jordan blo ck) is considered. Derogatory Jordan structures require further

investigation and new ideas. Diculties app earing in the derogatory case are asso ci-

ated with the lack of uniqueness of the transformation to a versal deformation and

will b e discussed in the conclusion.

The pap er is organized as follows. In section 2 some concepts of the singularity

COMPUTATION OF THE JORDAN NORMAL FORM 3

theory are intro duced and the general idea of the pap er is describ ed. Sections 3 and

4 give approximate formulae for matrices having a prescrib ed nonderogatory Jordan

structure in the neighb orho o d of a given generic (nondegenerate) complex or real

matrix A. In section 5 the same problem is studied in the case, when the initial matrix

A is degenerate. Conclusion discusses extension of the metho d to the derogatory case

and p ossibilities for its application to sp eci c typ es of matrices or matrix p encils.

In the pap er matrices are denoted by b old capital letters, vectors take the form

of b old small letters, and scalars are represented by small italic characters.

2. Bifurcation diagram. Let us consider an m m complex (real) nonsym-

metric matrix A holomorphically (smo othly) dep endentonavector of complex (real)

parameters p =(p ;:::;p ). The space of all m m matrices can b e considered as a

1 n

sp ecial case, where all entries of A are indep endent parameters.

Matrices A and B are said to have the same Jordan structure if there exists one-

to-one corresp ondence b etween di erent eigenvalues of A and B such that sizes of

Jordan blo cks in the JNFs of A and B are equal for corresp onding eigenvalues (JNFs

of the matrices A and B di er only byvalues of their eigenvalues). For example,

among the following matrices

1 1 0 1 0 0

2 1 0 3 0 0 5 1 0

A A @ A @ @

(2.1) ; C = 0 2 0 ; B = 0 2 1 A = 0 5 0

0 0 2 0 0 2 0 0 2

the matrices A and B have the same Jordan structure, while the matrix C has

a di erent Jordan structure. In the generic case a set of values of the vector p

corresp onding to the matrices A(p)having the same Jordan structure represents a

smo oth submanifold of the parameter space [1, 2]. This submanifold is called a stratum

(also called a colored orbit ofamatrix).Abifurcation diagram of the matrix family

A(p) is a partition (strati cation) of the parameter space according to the Jordan

structure of the matrix A(p)[1, 2]. Generic (typical) matrices A form a stratum of

zero co dimension consisting of nondegenerate matrices having only simple eigenvalues.

The bifurcation diagram has singularities at b oundary p oints of the strata (b oundary

of a stratum consists of strata having higher co dimensions). Due to singularities the

bifurcation diagram has rather complicated structure.

In this pap er we consider matrices having a nonderogatory multiple eigenvalue

(there is one eigenvector and, hence, one Jordan blo ck corresp onding to ). The strata

corresp onding to such matrices are denoted by , where d is the algebraic multiplicity

of the multiple eigenvalue . If the matrix A has several multiple nonderogatory

eigenvalues ;:::; with algebraic multiplicities d ;:::;d , then the corresp onding

1 s 1 s

stratum is denoted by . In the case of real matrix families we will di er

strata determined by real eigenvalues and complex conjugate pairs of eigenvalues by

d d

using notations and ( i! ) resp ectively.

Geometry of the strati cation can b e illustrated by the following example

2p p

1 2

(2.2) A(p)= ; p =(p ;p ;p ):

1 2 3

p 0

2 2

The bifurcation diagram consists of the following strata: (the cone surface p +

p p = 0 except the origin), (the origin), and the other part of the parameter space

2 3

corresp onding to matrices A(p) with distinct eigenvalues (interior and exterior of the

cone); see Figure 2.1. Here denotes the stratum corresp onding to a derogatory

4 ALEXEI A. MAILYBAEV

Fig. 2.1. Geometry of the bifurcation diagram.

double eigenvalue (with two corresp onding eigenvectors). The strata and are

asso ciated with a double eigenvalue, but di er byanumb er of the Jordan blo cks

(equal to the numb er of eigenvectors). Clearly, the strata and are smo oth

submanifolds of co dimension 1 and 3 resp ectively.

A matrix with multiple eigenvalues can b e made nondegenerate by an arbitrarily

small p erturbation. That is whyinnumerical analysis we usually deal with nonde-

generate matrices. A double nonderogatory eigenvalue represents the most generic

2 2

typ e of multiple eigenvalues. The corresp onding stratum ( in the real case) is a

smo oth hyp ersurface in the parameter space. Nonderogatory eigenvalues are the most

generic (typical) b etween all typ es of eigenvalues of given multiplicity. Nonderoga-

tory eigenvalues of multiplicity d determine the stratum of co dimension d 1 (the

has the co dimension d + + d s). In the real case the strata stratum

1 s

d d

and ( i! ) have the co dimension d 1and2(d 1) resp ectively [1,2,10].

Let us describ e an idea of the metho d used in this pap er. The versal deformation

theory allows studying di erent strata indep endently.From one side, this simpli es

the analysis, since a stratum is a smo oth manifold without singularities. From the

other side, we get additional information ab out the Jordan structure of a matrix.

b b

Let p be a point on the stratum ; see Figure 2.2. The neighborhood of p can b e

explored by means of a versal deformation. As a result, the stratum is describ ed

lo cally by the equation

0 0

(2.3) A (q (p)) = J ;

0 0

where A (q ) is a simple explicit family of d d matrices (a blo ckoftheversal

deformation); q (p)isa vector smo othly dep endentonp; J represents the Jordan

blo ck of dimension d. Let a p oint p of the parameter space b e given. Assuming that

the p oint p is close to p,we can write equation (2.3) in the form

@ A

0 0 0 2 0

A (q (p )) + A + O (kp p k )=J ; A = (2.4) (p p );

0 0 d i 0i

i=1

1=2

where kpk =(p p + + p p ) is the norm in the parameter space; derivatives

1 n

0 0 0

are calculated at p . The matrices A (q (p )) and @ A =@ p can b e found from the

0 0 i

analysis of the versal deformation at p . The latter represents the most nontrivial

)in(2.4),we obtain the equation determining part of the study. Omitting O (kp p k

a plane ,which is the rst order approximation of the stratum ; see Figure 2.2. A

distance from p to can b e found approximately as a distance to .Further analysis 0

COMPUTATION OF THE JORDAN NORMAL FORM 5

Fig. 2.2. Approximation of the stratum from the point p .

of the versal deformation allows nding approximations of multiple eigenvalues and

the Jordan chains (generalized eigenvectors) on the stratum. Note that all these

approximations use only information at the p oint p ,typically corresp onding to a

nondegenerate matrix A(p ).

The describ ed approach is similar to Newton's metho d for nding a ro ot of a

smo oth function, where the approximation of the ro ot is determined using the linear

approximation of the function at the initial p oint. Hence, if " =dist(p ; )isthe

d d

distance from p to , then the prop osed metho d gives the nearest p ointof with

2 d

the accuracy O (" ). Making k iterations, one nds a p oint of the stratum with the

accuracy O (" ).

3. Nonderogatory eigenvalue of a complex matrix. In this section we

study complex matrices A(p) holomorphically dep endentonavector of complex

parameters p. Let us consider a p oint p in the parameter space, where the matrix

b b

A = A(p) has a nonderogatory eigenvalue with algebraic multiplicity d. The corre-

b b

sp onding Jordan chain u ;:::;u (the eigenvector and asso ciated vectors, also called

1 d

generalized eigenvectors) is determined by the equations

b b

A u = u ;

1 1

b b b

A u = u + u ;

2 2 1

(3.1)

b b b

Au = u + u :

d d d1

b b

These vectors form an m d matrix U =[u ;:::;u ] satisfying equation

1 d

1 0

C B

C b b b b b B

(3.2) ; AU UJ =0; J =

C B

A @

b b

where J is the Jordan blo ck of dimension d. The left Jordan chain v ;:::;v corre-

1 d

sp onding to the eigenvalue (the eigenvector and asso ciated vectors of the transp osed

6 ALEXEI A. MAILYBAEV

matrix A ) is determined by the equations

T T

b b

v A = v ;

1 1

T T T

b b

b b b

v A = v + v ;

2 2 1

(3.3)

T T T

b b b

v A = v + v ;

d d d1

T T

b b b b

(3.4) v u =1; v u =0;i=2;:::;d:

d d

1 i

Equalities (3.4) represent normalization conditions uniquely determining the vectors

b b b b

v ;:::;v for given vectors u ;:::;u . System (3.3), (3.4) can b e written in the

1 d 1 d

equivalent matrix form

T T T

b b b b b b b

b b

(3.5) U = I; V =[v ;:::;v ]; A JV =0; V V

d 1

where I is the identity matrix. Note that the matrix V consists of the vectors

b b

v ;:::;v listed in the reversed order.

1 d

Let us consider a p oint p in the vicinityof p. In this section we consider the

generic case, when the matrix A = A(p ) has simple eigenvalues ;:::; with

0 0 1 d

the corresp onding eigenvectors u ;:::;u suchthat ;:::; ! as p ! p.The

1 d 1 d 0

left eigenvectors corresp onding to ;:::; are denoted by v ;:::;v . The m d

1 d 1 d

matrices U =[u ;:::;u ]andV =[v ;:::;v ] satisfy the equations

1 d 1 d

T T T

(3.6) A U UJ =0; V A J V =0; V U = I;

0 0 0 0

where the diagonal d d matrix J =diag( ;:::; ). The last equality of (3.6) is the

0 1 d

normalization condition, which uniquely de nes V for given U. Since the eigenvalue

b b

u as is nonderogatory (has one eigenvector u ), every eigenvector u tends to c

1 1 i i

p ! p for some constant c [23].

0 i

The goal of this pap er is to nd the rst order approximations (up to the terms

2 d

O (kp p k )) of the stratum in the parameter space, the nonderogatory multiple

eigenvalue on the stratum, and the corresp onding Jordan chain U.We are going to

nd these approximations using only information at the p oint p , where the matrix

A has simple eigenvalues. Toformulate the main result, let us intro duce the following

scalars and vectors of dimension n

=( + + )=d; = ; = ( ) ;

0 1 d i i 0 i i j

j =1;j6=i

(3.7)

@ A @ A

T T

u ;:::;v u ; i =1;:::;d; n = n =d; n = v

i i i i

i i

@p @p

1 n

i=1

where derivatives are taken at p . Let us de ne d d matrices K = diag(k ;:::;k ),

0 1 d

Y , and S, whose entries k , y , and s , i; j =1;:::;d, are

i ij ij

0 0 d T

; u ); u = u u u ; y = k =1=(u

1 1 ij i i

j i

(3.8)

s =(1) ; r = d j; s = :

ij i id i i i

r 1

1i <

1 r

i ;:::;i 6=i

1 r

COMPUTATION OF THE JORDAN NORMAL FORM 7

The matrix S can also b e expressed as S = R , where R is a d d matrix with the

(we assume that =0 even if = 0). The ith rowandj th comp onents r = 

j ij

j j

column of a matrix will b e denoted by S and S resp ectively.

Theorem 3.1. Let ;:::; be simple eigenvalues of the matrix A .Thenthe

1 d 0

rst order approximation of the stratum (where the eigenvalues ;:::; form a

1 d

nonderogatory eigenvalue of multiplicity d) is given by the fol lowing system of d 1

linear equations

0 T

q + rq (p p ) =0; j =1;:::;d 1;

j 0

d d

X X

(3.9)

0 d

q = s ; rq = (n n):

ij j i

j i

i=1 i=1

The rst order approximations of the multiple eigenvalue and the corresponding

b b

Jordan chain U =[u ;:::;u ] on the stratum are given by

1 d

(3.10) = +; = n(p p ) ;

0 0

U = U(K + K )+W S;

T 1

W =(A I v v ) (UKY +UK AUK) ;

0 i i

0 0 0 0 T

(3.11)

K =diag(k ;:::;k ); k = v WS = ; i =1;:::;d;

1 d i i

@ A

A = (p p );

i 0i

i=1

where al l derivatives are taken at p .

Let us write equations (3.9) in the matrix form

3 2 3 2

q rq

7 6 7 6

. .

(3.12) ; ; x = Q(p p ) = x; Q =

5 4 5 4

. .

q rq

where Q is a (d 1) n matrix; x is a vector of dimension d 1.

Corollary 3.2. Under the conditions of Theorem 3.1 the rst order approxi-

mation of the point p 2 nearest to p (a distanceismeasured by the Euclidean

min 0

norm) has the form

T T 1

p = p + x ( (3.13) QQ ) Q:

min 0

Similarly, Theorem 3.1 can b e used for nding the nearest p oint p, where the

matrix A(p) has a sp ecial multiple eigenvalue, for example, nonderogatory multiple

eigenvalue with the zero real part or absolute value equal to one (this is imp ortant

in stability problems). Theorem 3.1 can also b e used in the case of several di erent

nonderogatory eigenvalues.

Corollary 3.3. Let us consider a stratum , which is determinedby 

b b

matrices having s di erent nonderogatory eigenvalues ;:::; with multiplicities

1 s

d ;:::;d respectively. Then the rst order approximation of this stratum from the

1 s

b b

point p is the intersection of planes (3.9) calculated for al l ;:::; . The multi-

0 1 s

ple eigenvalues and the corresponding Jordan chains areapproximated by (3.10) and

(3.11) for every . i

8 ALEXEI A. MAILYBAEV

3.1. Entries of a matrix as indep endent parameters. If the space of all

complex m m matrices is considered, we can take all entries of A as indep endent

parameters. In this case the matrix A can b e used instead of the parameter vector

p. Then the derivative @ A=@ p reduces to a matrix with the unit on the (j; l )th

place and zeros on the other places, which corresp onds to the derivativeofA with

resp ect to its element a . The pro duct rq (p p ) in (3.9) takes the form of

jl j 0

trace Q (A A ) , where Q is an m m matrix corresp onding to rq ;traceA

j 0 i i

is a sum of elements standing on the main diagonal of the matrix. The norm kpk

1=2

transforms to the Euclidean matrix norm kAk = trace (A A) .

Let us intro duce the m m matrices

N = v u ; i =1;:::;d; N = (3.14) N =d:

i i i

i=1

These matrices are obtained from the vectors n , n (3.7) after substituting the pa-

rameter vector p by the matrix A and taking the derivative @ A=@ p in the form of

the matrix, which has the unit on the (j; l )th p osition and zeros on the other places.

Then Theorems 3.1, 3.3, and Corollary 3.2 can b e written in terms of matrices (3.14)

for the sp ecial case under consideration.

Theorem 3.4. Let ;:::; be simple eigenvalues of the matrix A .Thenthe

1 d 0

rst order approximation of the stratum (where the eigenvalues ;:::; form a

1 d

nonderogatory eigenvalue of multiplicity d)inthespaceofcomplex matrices is given

by the fol lowing system of d 1 linear equations

0 T

q + trace Q (A A ) =0; j =1;:::;d 1;

j 0

d d

X X

(3.15)

d 0

(N N): ; Q = = s q

i j ij

i j

i=1 i=1

The rst order approximations of the multiple eigenvalue and the corresponding

b b

Jordan chain U =[u ;:::;u ] on the stratum are given by

1 d

(3.16) = +; = trace N(A A ) ;

0 0

U = U(K + K )+W S;

1 T

W =(A I ) (UKY +UK (A A )UK) ; v v

(3.17)

0 i 0 i

T 0 0 0 0

= ; i =1;:::;d: WS = v ); k ;:::;k =diag(k K

i i d 1

Corollary 3.5. Under the conditions of Theorem 3.4 the rst order approx-

imation of the matrix A 2 nearest to A (with a distancemeasuredbythe

min 0

Euclidean matrix norm) has the form

A = A + (3.18) Q y ; y = P x;

min 0 j

j =1

where the (d 1) (d 1) matrix P has the elements p = trace (Q Q ); x and y

and y respectively. arecolumn-vectors of dimension d 1 with components x = q

j j

Note that the ab ove approximations use only information on simple eigenvalues

and corresp onding eigenvectors of the matrix A . 0

COMPUTATION OF THE JORDAN NORMAL FORM 9

3.2. Pro of. According to the versal deformation theory [1, 2], a family of ma-

trices A(p) in the vicinity of the p oint p 2 can b e represented in the form

(3.19) A(p)=C(p)A (q(p))C (p);

where C(p)isanm m nonsingular matrix smo othly dep endentonp; q(p)isa

0 0 00 00

e b e

vector smo othly dep endentonp suchthatq(p)=0;A(q ) = diag(A (q ); A (q )) is

a blo ck-diagonal matrix family with

0 1

q 0 0 0

B C

0 0 0

(3.20) A (q )=J + ; q =(q ;:::;q ):

B C

1 d

@ A

0 0 q 0

q q q

1 2 d

b b

Here J is the Jordan blo ck of dimension d with the eigenvalue (3.2). The d d

0 0 00 00

blo ck A (q ) corresp onds to the nonderogatory eigenvalue , while the blo ck A (q )

0 00

corresp onds to other eigenvalues; q =(q ; q ). According to (3.19), the matrices

0 0

A(p) and A(q(p)) have the same JNF. Note that the blo ck A (q )canbechosen in

many di erentways, and the sp ecial choice (3.20) is made in order to simplify the

calculations. The matrix family C(p) is uniquely determined on any smo oth surface

T (in the space of m m matrices), which is transversal to the plane

e b e b

(3.21) P = fB 2 C : AB BA(0) = 0g; A = A(p);

e e e

at C(p) 2 P and dim T +dimP = m [1, 2].

Multiplying (3.19) by C(p) from right, weobtainA(p)C(p)=C(p)A (q(p)).

Due to the blo ck-diagonal structure of A this equation splits into a set of indep endent

0 0

equations for the blo cks of A. The equation corresp onding to the rst blo ck A (q )

takes the form

0 0 0 0

(3.22) A(p)C (p)= C (p)A (q (p));

where C (p)isanm d matrix consisting of the rst d columns of C(p). Using the

e e

condition C(p) 2 T and the blo ck-diagonal structure of A(0) in (3.21), we nd that

the family C (p) is uniquely determined on any surface T of dimension d(m 1),

which is transversal (in the space of m d matrices) to the plane

0 md 0 0

b b b b b

(3.23) J =0g = fUX : X 2 cent Jg: P = fB 2 C : AB B

b b b

Here equality (3.2) was used. The centralizer cent J = fX 2 C : JX XJ =0g

consists of the matrices

1 0

x x x

1 2 d

C B

. .

C B

. .

0 x

(3.24) ; X =

C B

A @

0 0 x

0 0 0 x

where every upp er diagonal is lled by equal numb ers; the other places are all ze-

ros [11].

10 ALEXEI A. MAILYBAEV

Equation of the stratum has the form

(3.25) q (p)= = q (p)=0;

1 d1

which is equivalentto

0 1

 1 0 0

B C

0 0 

0 0

B C

A (q (p)) = J = (3.26) :

B C

@ A

0 0 1

0 0 0 

Here = + q (p) is an arbitrary constant determining the d d Jordan blo ck J .

d d

We assume that the p oint p b elongs to the vicinityof p, where (3.22) holds.

Then the functions q (p) can b e represented in the form

0 T 2

q (p)=q + rq (p p ) + O (kp p k );

i i 0 0

 

(3.27)

@q @q

i i

;:::; ; i =1;:::;d; q = q (p ); rq =

i 0 i

@p @p

1 n

where derivatives are evaluated at p . Using (3.27) in (3.25), we nd the rst order

approximation of the stratum as follows

0 T

(3.28) q + rq (p p ) =0; i =1;:::;d 1:

i 0

Equation (3.28) determines a plane of co dimension d 1 in the parameter space;

see Figure 2.2.

According to the assumptions of Theorem 3.1, the matrix A has simple eigen-

values ;:::; which coincide as p ! p. Hence, ;:::; are eigenvalues of the

1 d 0 1 d

0 0 0

= A (q (p )). Using (3.20) and comparing the characteristic equation for matrix A

d 0 d2 0 0 0

 q q q =0; = q ;

d1 2 1 d

with ( ) ( )= 0,we obtain

1 d

 + q = ;

(3.29)

r +1 0

= (1) ; r = d j +1; j =1;:::;d 1: q

t t

j 1 r

1t <

1 r

0 0 0

Expressions (3.20), (3.29) determine the matrix A = A (q (p )). Eigenvectors of

0 0

A form the d d matrix R satisfying equation A R = RJ , J =diag( ;:::; ).

0 0 1 d

0 0

Using (3.20), the matrix R can b e found explicitly as follows

0 1

1 1 1

B C

 

1 2 d

B C

2 2 2

B C

 

1 2

(3.30) : R =

B C

. . . .

@ A

. . . .

d1 d1 d1

 

1 2 d

COMPUTATION OF THE JORDAN NORMAL FORM 11

The matrix A can b e expressed in the form

0 1

(3.31) A = RJ S; S = R :

Elements of the matrix S are given in (3.8), what can b e veri ed bychecking the

relation SR = I. Using (3.20) in (3.31), we can rewrite expressions (3.29) in the form

0 0 d

(3.32) + q = ; q = R J S = s ; j =1;:::;d 1:

0 0 ij

d j i

i=1

Evaluating (3.22) at p ,multiplying it by R from the right side, and using ex-

pression (3.31), we nd

0 0 0 0

(3.33) A (C R) (C R)J =0; C = C (p ):

0 0 0

Hence, the ith column of the matrix C R is the eigenvector u multiplied byan

arbitrary constant k , and weobtain

C (3.34) = UKS; K = diag(k ;:::;k ):

1 d

We will sp ecify the matrix K later together with the calculation of the Jordan chain.

0 0 0

The functions A (q (p)) and C (p) can b e expressed in the form

0 0 0 0 2

A (q (p)) = A +A + O (kp p k );

(3.35)

0 0 0 2

C (p)= C +C + O (kp p k );

where

n n

0 0

X X

@ A @ C

0 0

A = (3.36) (p p ); C = (p p );

i 0i i 0i

@p @p

i i

i=1 i=1

derivatives are evaluated at p . Substituting (3.35) into (3.22), we obtain the equation

for the rst order terms

@ A

0 0 0 0 0 0

(3.37) (p p ): A C C A = C A AC ; A =

i 0i 0

0 0 0

i=1

1 T

Multiplying (3.37) by K V and R from left and right resp ectively, and using

expressions (3.6), (3.31), and (3.34), we nd

1 T 0 1 T 0 0 1 T

(3.38) J (K V C R) (K V C R)J = SA R K V AUK:

0 0

Taking the trace of (3.38), the left-hand-side vanishes and we obtain

T T

(3.39) rq (p p ) = n(p p ) =;

d 0 0

where the equality

0 0 0 T

trace (SA R) = trace (A RS) = trace (A )=drq (p p )

d 0

was used. Expressions (3.29), (3.39) give the rst order approximation (3.10) of the

2 T

multiple eigenvalue = + q (p)= + + O (kp p k ), = n(p p ) ,on

d 0 0 0

the stratum .

12 ALEXEI A. MAILYBAEV

Fig. 3.1. Geometry of the Jordan chain in the space C .

Taking the (i; i)th element of (3.38), the left-hand-side vanishes and weget

0 T T

(3.40) S A R = v Au = n (p p ) :

i i 0

Calculating the value of S A R with the use of (3.20) and (3.39), we obtain

T T

(3.41) s r rq (p p ) =(n n)(p p ) ; i =1;:::;d:

id ki k 0 i 0

k =1

Multiplying (3.41) by s =s = s = , summing over i from 1 to d, and using relations

ij id ij i

RS = I,we nd

T T

rq (p p ) = (3.42) (n n)(p p ) :

j 0 i 0

i=1

Substituting (3.32) and (3.42) into (3.28), we obtain the rst order approximation of

the stratum in the form (3.9).

b b

Now let us nd the approximation of the Jordan chain U =[u ;:::;u ] at p oints

1 d

of plane (3.9). Note that the Jordan chain U is not uniquely determined. All the

Jordan chains de ne the plane P (3.23) of dimension d in the space of m d matrices,

where any p oint of the plane P (except the p oints with det X = 0) represents the

Jordan chain. At the same time a set of matrices C (3.34) determines another plane

P of dimension d

(3.43) P = fUKS : K =diag(k ;:::;k )g;

1 d

The planes P and P are sketched on Figure 3.1 (the drawing is very schematic, since

the planes P and P usually have only one jointpoint at the origin).

We are going to nd the rst order approximation of the Jordan chain U (or,

equivalently, of the plane P ) using information at the p oint C 2 P . Hence, weneed

to take the p oint C , which is close to P , in order to get a b etter approximation.

0 0

Note that we seek a p oint C 2 P close to P keeping the norm of C b ounded

0 0

(kC k c>0), b ecause the planes P and P intersect at the origin. Since ! 0

b b

as p ! p, the limit of the matrix UK = C R as p ! p is a matrix with equal

0 0

b b

columns (see the form of the matrix R in (3.30)). At the same time C ! U 2 P

b b

= UKS 2 P is close to P , then the as p ! p. Hence, if p is close to p and C

0 0 0

COMPUTATION OF THE JORDAN NORMAL FORM 13

matrix UK has almost equal columns. This condition provides a wayforthechoice

of the appropriate matrix K. Its elements can b e taken, for example, in the form

(3.8). Expressions (3.8) for the elements k of the matrix K mean that the values

of pro jections of the vectors k u = UK on the direction u areequalto1.The

i i

vector u can b e chosen in di erentways. In particular, it can b e equal to the direction

determined by the eigenvector u (3.8).

The matrix C = UKS represents the zero-order approximation of the Jordan

1 T

chain U. Analogously, the matrix VK R represents the zero-order approximation

1 T

b b b b

of the left Jordan chain V , i.e., UKS ! U 2 P and VK R ! V as p ! p.The

matrix C already represents a go o d approximation of the Jordan chain; it has the

accuracy O (kp p k), which is enough for many applications. Now let us determine

the rst-order approximation of the Jordan chain (up to O (kp p k )) sp eci ed for

each p oint of the stratum .

The matrix family C (p) is uniquely determined on any surface T of dimension

b b

m(d 1) (in the space of m d matrices) transversal to the plane P at U = C (p);

see Figure 3.1 [1,2]. The surface T can b e chosen in the form

0 md T

b b

(3.44) T = C (p)+ fB 2 C : v B =0; j =1;:::;dg:

It is clear that dim T = m(d 1). The intersection of T and P is one p oint C (p),

since the condition v B =0 xthenumbers x standing on the corresp onding

diagonal of X; see expressions (3.23) and (3.24). Hence, T and P are transversal. Let

us consider a plane

0 0 md 1 T

T = C (3.45) + fB 2 C : R K V B =0; j =1;:::;dg

0 0 1 T

b b

obtained from (3.44) if we substitute C (p)by C and v by(VK R ) .

0 0 1 T 0

b b

Since C ! C (p) and VK R ! V as p ! p, the plane T is a small p erturbation

of T for p close to p; see Figure 3.1. Hence, T is transversal to P and we can uniquely

0 0 0 0 1 T 0

cho ose C (p) 2 T . The condition C (p) 2 T yields R K V C = 0 for

j =1;:::;d. These equalities can b e written in the equivalent form as follows

T 0

v (3.46) C =0; j =1;:::;d:

Multiplying (3.37) by R from right and using expressions (3.31), (3.34), we obtain

0 0 0

(3.47) A C R C RJ = UKSA R AUK:

0 0

Taking the ith column of (3.47), we nd

0 0

(3.48) (A I)(C R) =(UKSA R AUK) :

0 i

The matrix A I is singular with the simple zero eigenvalue. The solution

0 i

(C R) of (3.48) exists if and only if [27]

0 T

(3.49) (UKSA R AUK) =0: v

This condition is satis ed, since

T 0 0 T

v (UKSA R AUK) = k (S A R v Au )= 0;

i i

14 ALEXEI A. MAILYBAEV

where expression (3.40) was used. Then, the general solution (C R) has the

form [27]

0 0

(C R) = W + k u ;

(3.50)

T 1 0

W =(A I v v ) (UKSA R AUK) ;

0 i i

T 0

where A I v v is a nonsingular matrix; k is an arbitrary constant. Hence,

0 i i

i i

0 0 0 0 0

(3.51) C =(W + UK )S; K = diag(k ;:::;k ):

1 d

The matrix SA R in (3.50), calculated on plane (3.9), with the use of the explicit

forms of the matrices S and R (3.8), (3.30) is written as follows

(3.52) SA R = Y +I;

where elements of the matrix Y are determined in (3.8). Substituting expression

(3.51) into conditions (3.46), we nd

T 0 T 0

v C = v WS + k s =0;

j j j

T 0

WS = : = v k

j j

b b

Thus, we obtain the rst order approximation of the Jordan chain [u ;:::;u ]=

1 d

0 0 0 2

C (p)= C +C + O (kp p k ) in the form (3.11) at any p oint of plane (3.9).

This completes the pro of of Theorem 3.1.

The nearest to p p oint p 2 can b e found using approximation (3.9) or its

0 min

matrix form (3.12). In this case p p is the normal vector to plane (3.9). Hence,

min 0

T T

p p = (3.53) rq y = y Q; y =(y ;:::;y ):

min 0 j j 1 d1

j =1

Substituting (3.53) into (3.12), weget

(3.54) QQ y = x:

Finding y from (3.54) and substituting it into (3.53), we obtain expression (3.13) for

the nearest p oint of the stratum.

4. Nonderogatory eigenvalues of real matrices. In this section real matri-

ces A(p) smo othly dep endent onavector of real parameters p are considered. In

this case there are two di erenttyp es of nonderogatory eigenvalues: real eigenvalues

and complex conjugate pairs of eigenvalues. These eigenvalues form smo oth mani-

folds (strata) in the parameter space. We will denote the stratum corresp onding to

a real eigenvalue by , and the stratum corresp onding to a complex conjugate pair

of eigenvalues by( i! ) , where d is the algebraic multiplicity of a nonderogatory

eigenvalue. Since versal deformations in the real and complex cases are very similar,

the results for real matrices can b e obtained after a small change from Theorems 3.1

and 3.4.

4.1. Stratum ( i! ) . Let us consider a p oint p in the parameter space, where

the matrix A = A(p) has a nonderogatory complex eigenvalue with multiplicity

d. The corresp onding right and left Jordan chains form complex m d matrices U

COMPUTATION OF THE JORDAN NORMAL FORM 15

b b

and V satisfying equations (3.2) and (3.5). The real matrix A has also a complex

conjugate eigenvalue with the corresp onding right and left Jordan chains forming

b b

matrices U and V.

Let us consider a p oint p in the vicinityofp. We assume that the matrix

A = A(p ) has simple complex eigenvalues ;:::; suchthat ;:::; ! as

0 0 1 d 1 d

b b

p ! p.Ifp is close to p, the eigenvalues ;:::; have the same sign of the

0 0 1 d

imaginary part equal to the sign of Im . The right and left eigenvectors u and v

i i

corresp onding to form complex matrices U and V satisfying equations (3.6).

A blo ck of the versal deformation, corresp onding to a complex nonderogatory

eigenvalue , in the case of the real matrix family A(p) has the form (3.20), where

q (p);:::;q (p) are complex-valued smo oth functions of p (Re q (p) and Im q (p) are

1 d i i

indep endent smo oth functions) [2,10]. Hence, this case is almost identical to the

case of complex matrix families considered in the previous section. The di erence is

only that the system q (p)= 0, i =1;:::;d 1, determining the stratum ( i! )

represents 2(d 1) real equations (for real and imaginary parts).

Theorem 4.1. Let ;:::; be simple complex eigenvalues of the real matrix

1 d

A , having the same sign of the imaginary part sign(Im ). Then the rst order

0 i

approximation of the stratum ( i! ) (where the eigenvalues ;:::; form a

1 d

nonderogatory complex eigenvalue of multiplicity d) is given by the system of 2(d 1)

linear real equations (3.9), where every equality represents two equations for real and

imaginary parts. The rst order approximations of the multiple eigenvalue and the

corresponding Jordan chain U on the stratum are given by (3.10) and (3.11).

Corollary 4.2. Under the conditions of Theorem 4.1 the rst order approxi-

mation of the point p 2 ( i! ) nearest to p has the form

min 0

0 T 0 0 T 1 0

(4.1) p = p + x (Q Q ) Q :

min 0

where Q is a 2(d 1) n real matrix with the rows Re rq and Im rq , j =1;:::;d 1;

j j

0 0

and x is a real column-vector of dimension 2(d 1) with the components Re q

Im q .

If the stratum ( i! ) in the space of real m m matrices is considered, we

can take all entries of A as indep endent parameters (the matrix A is used instead of

the vector of parameters p). In this case the relations of Theorem 4.1 take the form

(3.15), (3.16), (3.17), where system (3.15) consists of 2(d 1) real equations (d 1

equations for the real parts and d 1 equations for the imaginary parts). The rst

order approximation of the matrix A 2 ( i! ) nearest to A (with resp ect to

min 0

the Euclidean matrix norm) has the form

2(d1)

0 0 0 1 0

A = A + Q y ; y = P x ;

min 0

j j

(4.2)

j =1

0 0 0 0 0 0

x = Re q ;x = Im q ; Q =ReQ ; Q =ImQ ;

i i

2i1 i 2i i 2i1 2i

0 0 T 0

); x where the real 2(d 1) 2(d 1) matrix P has the elements p = trace (Q Q

i j

0 0 0

; and y and y are real column-vectors of dimension 2(d 1) with the comp onents x

j j

the m m complex matrices Q are determined in (3.14), (3.15).

4.2. Stratum . Let us consider a p oint p in the parameter space, where the

b b

real matrix A = A(p) has a nonderogatory real eigenvalue with multiplicity d.

b b

The corresp onding right and left Jordan chains form real m d matrices U and V satisfying equations (3.2) and (3.5).

16 ALEXEI A. MAILYBAEV

Let us consider a p oint p in the vicinityofp. We assume that the matrix

A = A(p ) has simple eigenvalues ;:::; suchthat ;:::; ! as p ! p.

0 0 1 d 1 d 0

The eigenvalues ;:::; are real or complex conjugate. Let u and v b e the right

1 d i i

and left eigenvectors corresp onding to and forming the matrices U and V that

satisfy equations (3.6); eigenvectors corresp onding to real eigenvalues are real and

eigenvectors corresp onding to complex conjugate eigenvalues are complex conjugate.

In the case of real matrix families a blo ckoftheversal deformation, corresp onding

to the real nonderogatory eigenvalue , has the form (3.20), where q (p);:::;q (p)

1 d

are real smo oth functions of p [2,10]. Thus, the case is almost identical to the

case for complex matrix families considered in the previous section. Note that

the Jordan chain U obtained by expressions (3.11) can b e complex, if the vector u

determining the matrix K is complex. To get the real Jordan chain wehavetocho ose

the real vector u for determining elements of the matrix K in (3.8). This can b e done

as follows

 . 

T 0 0

K =diag(k ;:::;k ); k =1=(u (4.3) u ); u =Re u u u :

1 d i 1 1

The matrix K (4.3) has the required prop erty: if the vectors u are close to c u with

i i 1

some constants c , then the columns of the matrix UK will b e close to each other.

With the use of conditions (4.3) the eigenvectors (UK) = k u , corresp onding to

i i

complex conjugate eigenvalues , are complex conjugate, and the eigenvectors k u ,

i i i

corresp onding to real eigenvalues , are real. Then it can b e shown that though the

matrices U, V and the eigenvalues can b e complex, equations (3.9) and the values

of and U found by (3.10) and (3.11) are real.

Theorem 4.3. Let ;:::; bereal or complex conjugate simple eigenvalues of

1 d

the real matrix A . Then the rst order approximation of the stratum (wherethe

eigenvalues ;:::; form a nonderogatory real eigenvalue with multiplicity d)is

1 d

given by the system of d 1 linear real equations (3.9). The rst order approximations

of the multiple real eigenvalue and the corresponding real Jordan chain U on the

stratum are given by (3.10) and (3.11), where the matrix K has the form (4.3).

Corollary 4.4. Under the conditions of Theorem 4.1 the rst order approxi-

mation of the point p 2 nearest to p has the form

min 0

T T 1

(4.4) p = p + x (QQ ) Q;

min 0

where the real matrix Q and the real vector x are given by (3.12).

If the stratum in the space of real m m matrices is considered, we can take

all entries of A as indep endent parameters (the matrix A is used instead of the vector

of parameters p). In this case relations of Theorem 4.3 take the form (3.15), (3.16),

(3.17) with the matrix K from (4.3). The rst order approximation of the matrix

A 2 nearest to A has the form (3.18).

min 0

Finally, let us consider the case, when the real matrix A has several di erent

nonderogatory multiple eigenvalues.

d d

t d 1

t+1

( i! ) ( Corollary4.5. Let us consider a stratum

t+1 t+1 s

i! ) , which is determined by matrices having t di erent nonderogatory real eigen-

values with multiplicities d ;:::;d and s t di erent pairs of complex conjugate non-

1 t

derogatory eigenvalues with multiplicities d ;:::;d . Then the rst order approxi-

t+1 s

mation of the stratum is the intersection of planes (3.9) calculatedbyTheorems 4.1

and 4.3 for every pair of complex conjugate multiple eigenvalues and every multiple

real eigenvalue. The multiple eigenvalue and the corresponding Jordan chain are given

COMPUTATION OF THE JORDAN NORMAL FORM 17

Fig. 4.1. Approximations of the nearest points of the stratum from di erent points p .

and ( i! ) (the matrix K for real multiple by (3.10) and (3.11) for every

j j

eigenvalues is taken from (4.3)).

4.3. Example 1. Let us consider a two-parameter matrix family

0 1

1 3 0

@ A

p 1 p

A(p)= (4.5) ; p =(p ;p ):

1 2

2 3 1

We will calculate eigenvalues of A = A(p ) at di erentpoints p . Then the nearest

0 0 0

to p p oint of the stratum is evaluated by Corollary 4.4. In this case (d =2)

system (3.9) represents a single linear equation

T 0

(4.6) + rq (p p ) =0; q

1 0

0 2

where q = , rq = (n n ), and =( )=2. Then the nearest to p vector

1 2 1 2 1 0

p 2 is approximated by

min

p = p (4.7) rq :

min 0 1

rq rq

In calculations wetake and equal to the complex conjugate pair of eigenvalues

1 2

of A . If all eigenvalues of A are real, we consider all three p ossibilities for ,

0 0 1

 and then takeapair , providing the minimal distance kp p k.The

2 1 2 min 0

result is shown on Figure 4.1. Here the arrows are the vectors p p , where p

0 min min

is the nearest p ointof found by (4.7) for di erent p . The bifurcation diagram

is represented by a solid line (it is found explicitly by calculating the discriminantof

the characteristic p olynomial of (4.5)). For one p oint p on Figure 4.1 two iterations

were p erformed, when p is taken as a new starting p oint p . The results show

min 0

high accuracy of the approximation.

Let us consider a p oint p =(0:3; 9:1), where the matrix A has eigenvalues

0 0

 = 2:624, = 1:472, = 7:096. Calculations with the use of (4.7) show

1 2 3

that the pair , gives the smallest distance to the stratum .Found by (4.7)

1 2

and Theorem 4.3 approximate values of the nearest p oint p 2 , the double min

18 ALEXEI A. MAILYBAEV

b b b

eigenvalue , and the corresp onding Jordan chain U =[u ; u ] (the eigenvector u

1 2 1

and the asso ciated vector u )atp are as follows

2 min

0 1

0:7044 0:1496

@ A

b b 0:7044 0:0852

p =(0:0008; 8:9990); = 2:00006; [u ; u ]= :

min 1 2

0:2346 0:1067

Here the matrices S, Y , and quantities , used in (3.11) are

1 2

 

1=2 1=(2 ) 1 1

S = ; Y = ; = 1=(2 ); = :

1 2 1

1=2 1=(2 ) 1 1

The distance from p to is equal to kp p k =0:317. For comparison, exact

0 min 0

b b

values of p , ,andu , u (found by the direct analysis of eigenvalues of matrix

min 1 2

family (4.5)) are

0 1

0:7044 0:1494

@ A

b b 0:7044 0:0854

p =(0; 9); = 2; [u ; u ]= :

min 1 2

0:2348 0:1067

Note that all the approximations were found using only simple eigenvalues and eigen-

vectors of the matrix A and derivatives of A(p) with resp ect to parameters at p .

0 0

If we consider the stratum in the space of real matrices (all entries of the

matrix A are parameters), then approximations of the matrix A 2 nearest to

min

the matrix A = A(0:3; 9:1) and the corresp onding Jordan chain are found by(3.18)

and (3.17) as follows

0 1 0 1

0:9774 3:0219 0:0065 0:6962 0:1480

@ A @ A

0:2886 1:0119 9:0962 b b 0:6961 0:0821

A = ; [u ; u ]= :

min 1 2

2:0345 2:9654 1:0107 0:2119 0:1088

In this case the distance to the stratum and the double eigenvalue of A found

min

by (3.16) are equal to kA A k =0:0618 and = 2:0459.

min 0 min

Near the origin p =0,which represents the stratum of higher co dimension ,

approximation (4.6) gives less accurate results. More generally, derived approxima-

tions are less accurate, when the distance to the stratum under consideration is

of the same magnitude as the distance to some less generic stratum (for example,

d 0

for d >d). The presence of a less generic stratum means that the p oint p is

near the b oundary of . Near the b oundary the size of the neighb orho o d, where

transformation to the versal deformation (3.22) holds, decreases. Nevertheless, we

always get a go o d approximation of the least generic stratum lying near p , since p

0 0

is not near its b oundary. This prop ertyisvery useful, b ecause using information on

the least generic stratum near p ,we can determine all the bifurcation diagram in the

vicinity [1, 2, 16]. On Figure 4.2 approximations (4.7) of the nearest p ointof are

shown by solid arrows for di erentpoints p near the origin ( ); for the same p oints

approximations (3.12), (4.4) for the nearest p oint of the stratum are shown by dot-

ted arrows. It can b e seen from Figure 4.2, that though approximations of the p oints

2 3 3

p 2 are less accurate for p near , approximations of the p oints p 2

min 0 min

provide very go o d results. Having information on the p oint p 2 (triple eigen-

min

value and the corresp onding Jordan chain), we can determine the bifurcation diagram

lo cally, in particular, approximate the cusp singularity of the bifurcation diagram for

the example under consideration [16].

COMPUTATION OF THE JORDAN NORMAL FORM 19

Fig. 4.2. Behavior of the approximation in presence of a less generic stratum.

4.4. Example 2. In this example we study accuracy of the approximations of

strata and convergence of the metho d, if wewant to nd the nearest matrix with the

prescrib ed Jordan structure.

Let us takea10 10 real matrix A equal to the JNF having a nonderogatory

eigenvalue = 2 with multiplicity 4 and simple eigenvalues = 4; 3; 2; 1; 0; 4.

We consider p erturbations of A in the form

A = A + D;

0 d

where D is a 10 10 real normally distributed random matrix having the xed norm

kDk =1=2. Using the nearest to = 2 simple eigenvalues ;:::; of the matrix A

1 4 0

in formulae (3.15), (3.18), we obtain the approximation A of the nearest matrix of

min

the stratum (in the space of real matrices) and the approximation of the distance

4 4

from A to in the form dist(A ; )=kA A k. The mean value of this

0 0 min 0

distance, obtained after 100 numerical exp eriments with di erent random p erturba-

tions D, is equal to 0:087. It is clear that the matrix A should b e di erent from

min

A and the distance from A to A should b e much less than 1=2 (the distance

d 0 min

to A ). Since the stratum has co dimension 3, the p erturbation D has, in the

average, 3 elements in the normal direction to , while other 97 elements determine

4 4

p erturbations parallel to . Hence, the mean value of the distance from A to

should b e 0:03 kDk 0:087, which con rms correctness of the ab ove results.

To determine accuracy of the approximation A ,we can nd a distance from

min

4 0

= A as a new starting p oint. As a result, we obtain A to bytakingA

min min

0 4 0 0

the distance dist(A ; )=kA A k giving the error of the approximation A .

min

0 0

min

00 0

Nowwe can take A = A as a new starting p oint etc. This sequence of matrices

0 min

converges to a p ointof very fast. The results of such iterations averaged over

100 exp eriments are shown in the rst rowofTable 4.1. It turns out that after the

third iteration we get a matrix with a nonderogatory real eigenvalue of multiplicity4

with the accuracy ab out 8:3 10 (the accuracy for the ith iteration is given in the

(i + 1)th column).

Similar calculations were carried out for the stratum ( i! ) , where a 10 10

real matrix A with a pair of nonderogatory eigenvalues =1 2i of multiplicity

4 and simple eigenvalues = 3; 1was considered. The results for 4 iterations

averaged over 100 random p erturbations D, kDk =1=2, are given in the second row

of Table 4.1. Here we calculated matrices A 2 ( i! ) having a pair of complex min

20 ALEXEI A. MAILYBAEV

Table 4.1

4 4

Iterative calculations of matrices of the strata and ( i! ) . Distances to the strata for 4

iterations averaged over 100 experiments with di erent perturbations A = A + D.

st nd rd th

1 iteration 2 iteration 3 iteration 4 iteration

4 2 3 7 15

dist(A ; ) 8:7 10 2:3 10 4:1 10 8:3 10

4 1 3 7 15

dist(A ; ( i! ) ) 1:2 10 2:8 10 3:4 10 3:4 10

conjugate nonderogatory eigenvalues of multiplicity 4 using expressions (4.2). One

can see that the convergence is very fast and 3 iterations give the matrix of ( i! )

with the accuracy ab out 3:4 10 .

5. Approximation of the stratum from a nongeneric p ointp. In

this section the case, when eigenvalues ;:::; of the matrix A are not distinct,

1 d 0

is considered. This is the nongeneric case and, hence, it is less imp ortant from the

practical p oint of view. Nevertheless, we should study this case to complete the

metho d for approximating matrices with nonderogatory eigenvalues. Results of this

section can b e useful, when we can not distinguish some of the eigenvalues among

 ;:::; and, therefore, have to treat some of them as multiple. We consider only

1 d

d n

complex matrix families (the stratum in the space of complex parameters C ).

Approximations for real matrix families can b e easily derived from the complex case,

as it was done in section 4. Since we use the same considerations based on the versal

deformation theory, only the main steps of the analysis will b e describ ed without

going muchinto details.

b b

Let b e a nonderogatory multiple eigenvalue of the matrix A = A(p) with

multiplicity d and let ;:::; b e eigenvalues of the matrix A = A(p ) suchthat

1 d 0 0

 ! as p ! p. According to the singularity theory [1, 2], for p oints p in

i 0 0

the vicinityofp eigenvalues ;:::; are simple or nonderogatory multiple. Let us

1 d

0 0

denote di erent eigenvalues among ;:::; by ;:::; and their multiplicities

1 d

by d ::: d , such that = = = , s =1;:::;K, where i =0,

1 K i +1 i +d 1

s s s s

i = d , ::: , i = d + ::: + d ; d = d + + d . The Jordan structure of A

2 1 K 1 K 1 1 K 0

corresp onding to the eigenvalues ;:::; is describ ed bythe d d matrix

1 d

1 0

 1

1 0

C B

s .

C B

(5.1) ; ; J = J =

A @

C B

A @

. K

s 0

where J is the d d Jordan blo ck corresp onding to the eigenvalue . The m d

s s

0 s

matrices U =[u ;:::;u ]andV =[v ;:::;v ], satisfying equations (3.6) with the

1 d 1 d

matrix J from (5.1), determine the right and left Jordan chains u ;:::;u

0 i +1 i +d

s s s

and v ;:::;v corresp onding to .

i +d i +1

s s s s

The matrix family A(p) in the vicinityofthepoint p 2 can b e expressed in

0 0

form (3.22), where A (q ) is the blo ckofversal deformation (3.20) corresp onding to

the nonderogatory multiple eigenvalue of multiplicity d.We assume that the p oint

0 0 0

= A (q (p )) p b elongs to the vicinityofp, where (3.22) holds. The matrix A

0 0

has the eigenvalues ;:::; and, hence, it is determined by expressions (3.20),

1 d

0 0 0

,we of A ;:::; (3.29). Finding Jordan chains corresp onding to the eigenvalues 

0 1 K

COMPUTATION OF THE JORDAN NORMAL FORM 21

determine a d d matrix R with the elements

0; i

; C = (5.2) r = ; i

it t1 ij s s s s

; i t; C

t!(i t)!

j i1

0 0

where = , =( + + )=d =(d + + d )=d. The matrix R

j j 0 0 1 d 1 K

0 0

satis es the equation A R = RJ and the matrix A is expressed in the form (3.31),

0 0

where S = R for the new matrix R (5.2). Using (3.31), we can rewrite expressions

(3.29) for the quantities q as follows

0 0

(5.3) + q = ; q = R J S ; j =1;:::;d 1:

0 0

d j

The matrix C can b e found from (3.33) as follows

C (5.4) = UKS; K 2 cent J ;

where K 2 cent J is a d d matrix commuting with J and having the form [11]

0 0

0 1

k k k

i +1 i +2 i +d

s s s s

B C

. .

B C B C

. . .

0 k

i +1

K = ; K = (5.5) :

B C

@ A

B C

@ A

k 0 0

i +2

0 0 0 k

i +1

The d d matrix K has equal numbers k on the j th upp er diagonal and zeros

s s s i +j

b elow the main diagonal.

0 0

For the rst order terms A and C (3.36) wehave equation (3.38). Taking

the sum of (i; j )th elements of (3.38), where i = j + d t, j = i +1;:::;i + t,

s s s

and using (3.20), (5.1), (5.5), (5.10), the left-hand-side vanishes and we obtain the

equation

0 T 1 T T

(5.6) trace (SA RT ) = trace (K V AUKT );

where T is a d d matrix having units on the (i; j )th p ositions for i = j + d t,

j = i +1;:::;i + t, and zeros on the other places. Since the matrices T and K

s s

commute for any K 2 cent J ,wehave

1 T T T T

(5.7) trace (K V AUKT )=trace(V AUT ):

With the use of expression (5.7) equation (5.6) takes the form

T T

(5.8) f rq (p p ) = m (p p ) ;

zl l 0 z 0

l=1

1 n

where z = i + t 2f1;:::;dg. Elements of the vectors m =(m ;:::;m ) and the

s z

z z

scalars f are as follows

@ A

l T

m = v u ;

z j +d t

j =i +1

s r ; i

> (5.9)

lj s s s s

(j +d t)d

j =i +1

f =

0; i

s s s s

d ; z = i + d ;l= d;

s s s

22 ALEXEI A. MAILYBAEV

where derivatives are evaluated at p .

Taking the sum of equations (5.8) for z = i + d , s =1;:::;K,we obtain

s s

T T

(5.10) = rq (p p ) = m(p p ) ;

d 0 0

where

1 @ A

1 n l T

m =(m ;:::;m ); m = v (5.11) u :

d @p

j =1

Expression (5.10) determines the approximation of a multiple eigenvalue on the

stratum in the form

(5.12) = +:

If we substitute the value of rq (p p ) from (5.10) into (5.8) and omit the

d 0

equation for z = d, whichisnow linearly dep endent on the others, we nd a system

of d 1 linear equations determining the quantities rq (p p ) , j =1;:::;d 1,

j 0

as follows

T T

(5.13) f rq (p p ) =(m f m)(p p ) ; i =1;:::;d 1:

ij j 0 i id 0

j =1

Solving (5.13) with resp ect to rq (p p ) ,we nd

j 0

T T

(5.14) rq (p p ) = h (m f m)(p p ) ; j =1;:::;d 1;

j 0 ji i id 0

i=1

where H = F with H and F b eing (d 1) (d 1) matrices, whose elements are

h and f , i; j =1;:::;d 1.

ij ij

Expressions (5.3) and (5.14) determine system of d 1 linear equations

0 T

(5.15) q + rq (p p ) =0; j =1;:::;d 1;

j 0

which is the linear approximation of the stratum .

Finally,we need to nd the approximation of the Jordan chain U on the stratum

d 0

b b

 . All the Jordan chains U form the plane P (3.23), and all the matrices C (5.4)

form the plane

(5.16) P = fUKS : K 2 cent J g:

0 0

To get a b etter approximation of U = C (p)we should take the matrix C 2 P ,

0 0

b b

whichisclosetoP . The matrix C R = UK converges to the matrix C (p)R as

p ! p, where nonzero elements of the d d matrix R are rb =1,t =1;:::;d ,

0 s

t(i +t)

s =1;:::;K; see expression (5.2) for elements of the matrix R. Hence, to get the

matrix C close to P we need to cho ose the matrix K 2 cent J such that the columns

i + t; i + t;::: (except the cases i + t>i + d ) of the matrix UK are close to each

1 2 s s s

b b

other. These columns converge to u as p ! p. The appropriate matrix K can b e

t 0

chosen in the form (5.5), where

 

0 0 T

u ); u = u u u ; k =1 (u

1 1 i +1

s i +1

(5.17) .

0 0 T T

);t=2;:::;d ;s=1;:::;K: u u k = k u (u

s i +t i +t+1z

s s i +z i +1

s s

z =2

COMPUTATION OF THE JORDAN NORMAL FORM 23

Expressions (5.17) mean that the vectors k u = UK have the unit

i +1 i +1

s s

pro jection on the direction u , while the vectors UK , t =2;:::;d ,havethe

zero pro jection on this direction.

The matrix C is determined by equation (3.47), which can b e written for each

column in the form

0 0 0 0

s s

(A I)(C R) =(C A R AUK) ;

s 0

0 0 0 0 0

(A I)(C R) (C R) =(C A R AUK) ;

(5.18)

s 0

t = i +2;:::;i + d ; s =1;:::;K:

s s s

A solution of system (5.18) exists for the p oints p on the plane (5.15) approximating

 . A general form of this solution is as follows [27]

0 0 0

C =(W + UK )S; K 2 cent J ;

1 0 0

s s

W = Z (C A R AUK) ;

s 0

1 0 0

W = Z (C A R AUK) + W ; (5.19)

s 0

0 T

Z = A I v v ;

s 0 i +d

s s s i +1

t = i +2;:::;i + d ; s =1;:::;K;

s s s

where Z is a nonsingular matrix (v is the eigenvector corresp onding to and

s i +d

s s s

0 0

v is the last vector in the Jordan chain of ); K is an arbitrary matrix of the

i +1

s s

form (5.5).

d 0

In the vicinityof the matrix family C (p) is uniquely determined on the surface

0 0 0

T (3.45) in the space of m d matrices. The condition C (p) 2 T leads to equations

(3.46). Substituting (5.19) into (3.46), we nd the matrix K in the form (5.5) with

the elements

0 T

k = v WS =s ;

(i +d )d

i +1 s s

i +d

s s

0 T 0

(5.20)

k = v WS + k s s ;

(i +d t+z )d (i +d )d

i +t i +d t+1 i +z

s s s s

z =1

t =2;:::;d ;

where s are elements of the matrix S. Expressions (5.4), (5.17), (5.19), and (5.20)

0 0

give the rst order approximation of the Jordan chain U = C +C on the stratum

 .

The results of this section can b e represented in the form of a theorem.

0 0

Theorem 5.1. Let ;:::; be simple or nonderogatory multiple eigenvalues of

the matrix A with multiplicities d ;:::;d respectively; d = d + + d . Then the

0 1 K 1 K

d 0 0

rst order approximation of the stratum (where the eigenvalues ;:::; form a

nonderogatory eigenvalue of multiplicity d) is given by the system of d 1 linear

and rq determined in (5.1){(5.3), (5.11), (5.9), and (5.14). equations (5.15) with q

The rst order approximations of the multiple eigenvalue and the corresponding

0 0 d

Jordan chain U = C +C on the stratum are given by (5.10){(5.12) and (5.4),

(5.5), (5.17), (5.19), (5.20).

Corollary 5.2. Under the conditions of Theorem 5.1 the rst order approxi-

mation of the point p 2 nearest to p (a distanceismeasured by the Euclidean

min 0

norm) has the form

T T 1

( p = p + x ) (5.21) QQ Q;

min 0

24 ALEXEI A. MAILYBAEV

where the matrix Q and the vector x are determined in (3.12) with the quantities q

and the vectors rq determined by Theorem 5.1.

Note that Theorem 5.1 gives lo cal approximations of ,multiple eigenvalue ,

and the Jordan chain U even if p b elongs to . Theorem 3.1 is a sp ecial case of

0 0

Theorem 5.1, when all the eigenvalues ;:::; are simple. Similar results for real

matrix families can b e easily obtained from Theorem 5.1 in the same way,asitwas

done in section 4.

5.1. Example. Let us consider matrix family (4.5). At the p oint p =(2=3; 1=3)

the matrix A has a double nonderogatory eigenvalue = = = 0 and a simple

0 1 2

eigenvalue = =3(multiplicities are d = 2 and d =1). Hence, p is a

3 1 2 0

nongeneric p oint of the parameter space. Let us nd the approximation of the stratum

 using information at p . The matrices J and U =[u ; u ; u ], V =[v ; v ; v ]in

0 0 1 2 3 1 2 3

this case have the form

0 1 0 1 0 1

2=3 0 1=3 1 1 1 0 1 0

@ A @ A @ A

; ; V = 5=3 1 2=3 ; U = 1=3 0 2=3 0 0 0 J =

2=9 1=3 1=9 1 3 2 0 0 3

0 0

; u and v are right , where u , u and v , v are right and left eigenvectors of 

2 1 1 3 2 3

2 1

. The matrices R, S, and K havethe and left asso ciated vectors corresp onding to 

form

1 1 0 0

k k 0 1 0 1

1 2

A A @ @

0 k 0 1 

; ; S = R ; K = R =

1 1 3

2 2

0 0 k 2 

3 2

1 3

0 0 0 0

where = = = 1, = =2, =(2 + )=3 = 1. Us-

1 2 0 3 0 0

1 2 1 2

ing expressions (5.3), (5.11), (5.9), and (5.14), we ndtwo equations (5.15) for the

approximation of the stratum as follows

2+ 6(p 1=3) = 0;

(5.22)

3+ 3(p 2=3)+3(p 1=3) = 0:

1 2

System (5.22) have a solution p =(0; 0), which is the approximation of . The triple

b b b b

nonderogatory eigenvalue and the Jordan chain U =[u ; u ; u ]atp calculated by

1 2 3

formulae (5.10){(5.12) and (5.4), (5.17), (5.19), (5.20) have the form

0 1

0 189 1

@ A

 =1; U = (5.23) 0 0 63 :

209

378 191 20

Direct calculations show that the p oint p =(0; 0), quantity = 1, and the matrix

U (5.23) represent the exact stratum ,multiple eigenvalue, and corresp onding Jor-

0 0

dan chain. The reason for such accuracy is that q (p)and C (p) are linear functions

determined in the whole parameter space

0 0 0 0 0

A(p)C (p)=C (p)A (q (p)); q (p)= (q ;q ;q )=(6p ; 3(p + p ); 0);

1 2 3 2 1 2

1 0

6p 1+2(p + p ) 1

2 1 2

A @

; C (p)= 2p p1+3p (1 + 2(p + p ))=3

2 2 1 2

2(1 p +2p ) 3+2(p + p ) 3

1 2 1 2

and Newton's metho d, whose idea was used for nding the approximations, gives exact solutions for linear functions.

COMPUTATION OF THE JORDAN NORMAL FORM 25

6. Conclusion. In this pap er a practical numerical metho d of nding non-

derogatory multiple eigenvalues and corresp onding Jordan chains of complex and

real matrices dep endentonseveral parameters is develop ed. Similar problem in the

space of all matrices is considered as a sp ecial case. The advantage of the suggested

approach is that given a p oint p in the parameter space we can approximate the

nearest p oint p 2 , where the matrix A(p ) has a nonderogatory eigenvalue

min min

with multiplicity d, using only information at p . Moreover, we can determine the

geometry of the stratum in the parameter space and nd approximations of the

multiple eigenvalue and the corresp onding Jordan chain at p oints of this stratum. The

approach is based on the versal deformation theory. As a result, it gives quantitative

information ab out the strati ed structure of the parameter space near the p oint p ,

which can not b e achieved by other metho ds. This information allows studying mul-

tiple eigenvalues of sp ecial form, like eigenvalues with a given real part or absolute

value, which is imp ortant in stability problems.

Multiple eigenvalues play signi cant role in problems of stability and dynamics,

since they are asso ciated with the singular b ehavior of a system. Therefore, it is im-

p ortanttoknow whether the system matrix has multiple eigenvalues. Assuming that

the vector of parameters p is given with the accuracy ",we need to nd a p oint p in

the "-vicinityof p such that the matrix A(p) has an eigenvalue of highest multiplicity

(more correctly, the matrix with the least generic Jordan structure). This represents

avery imp ortant problem of numerical calculation of multiple eigenvalues, since the

least generic Jordan structure determines lo cal characteristics of the sp ectrum [1,2].

The results of this pap er solve this problem in the part concerning nonderogatory

eigenvalues, i.e., we can approximate a p oint p, kp p k <", where the matrix A(p)

has a nonderogatory eigenvalue of the highest multiplicity.

Analysis of derogatory eigenvalues of matrices A(p) lead to additional diculties,

which require mo di cation of the metho d. These diculties are asso ciated with the

fact that a versal deformation is not universal in the derogatory case [1, 2]. This

means that the function q (p) is not uniquely determined by the matrix family A(p).

0 0

As a result, the matrices A and A can not b e found only from the analysis of the

0 0 0 0

equation A(p)C (p)=C (p)A (q (p)). Hence, it is necessary to use some additional

considerations on the geometry of the strati cation near p oints p, where derogatory

eigenvalues exist. This will b e done in the next pap er.

Application of the versal deformation theory to calculation of nongeneric nor-

mal forms can b e carried out in the same way for sp ecial typ es of matrices suchas

Hamiltonian or reversible matrices. Using versal deformations of these matrices in

the corresp onding matrix spaces, one can deriveapproximations of strata, multiple

nonderogatory eigenvalues, and corresp onding Jordan chains sp eci ed for di erent

typ es of matrices. These approximations can b e useful for the analysis of stability

and dynamics of sp ecial typ es of mechanical and physical systems. Similarly,versal

deformations can b e used for calculation of the nongeneric Kronecker canonical form

of a matrix p encil.

Acknowledgments. The author thanks Alexander P. Seyranian for suggesting

this problem and useful comments.

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