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SIAM J. MATRIX ANAL. APPL. Vol. 21, No. 2, pp. 396­417

c 1999 Society for Industrial and Applied Mathematics

TRANSFORMATION OF FAMILIES OF MATRICES TO NORMAL FORMS AND ITS APPLICATION TO STABILITY THEORY
ALEXEI A. MAILYBAEV Abstract. Families of matrices smo othly dep ending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29­43] and Galin [Uspekhi Mat. Nauk, 27 (1972), pp. 241­242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can b e transformed in the vicinity of a p oint in the parameter space by a change of basis, smo othly dep endent on a vector of parameters, and by a smooth change of parameters. In this pap er a constructive metho d of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with resp ect to parameters are determined from a recurrent pro cedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested metho d allows using efficiently miniversal deformations for investigation of different prop erties of matrix families. This is shown in the pap er where tangent cones (linear approximations) to the stability domain at the singular b oundary p oints are found. Key words. family of matrices, normal form, versal deformation, stability, tangent cone AMS sub ject classifications. 15A21, 15A90, 58F36 PI I. S0895479898338378

1. Intro duction. The paper deals with the study of families of complex and real matrices (a family of matrices is a matrix-valued function A(p) holomorphically (smoothly) depending on a vector p of complex (real) parameters). It is known that from all families of matrices we can select such families that transformation of any family to them can be carried out by a change of basis, smoothly depending on parameters, and by a smooth change of parameters [1, 2, 3]. Such families are called versal deformations. Versal deformations with the minimum possible number of parameters are called miniversal deformations or normal forms. Miniversal deformations of complex matrices have been found by Arnold [1, 2, 3]. He showed that the miniversal deformation, to which a family of matrices A(p) can be transformed in the vicinity of a point p = p0 , is determined by the Jordan form of the matrix A(p0 ). Miniversal deformations of real matrices have been studied by Galin [8]. to .5pt The important part of the transformation of a family of matrices to the normal form is to find functions describing a change of basis, depending on parameters, and a change of parameters. Determination of these functions is necessary for effective use of normal forms, but no general algorithm for finding them is available yet. Efforts of construction of the transformation functions required in a specific case have been done in [5, 12]. In the case when the matrix A0 = A(p0 ) has only nonderogatory eigenvalues, the problem of construction of transformation to the normal form has been discussed in [13]. However, the method of finding the transformation functions suggested in [13] can be used only for matrices of small dimensions or in the case when the matrix A0 has only one nonderogatory eigenvalue. Otherwise it leads to a
Received by the editors May 6, 1998; accepted for publication (in revised form) by M. Overton February 22, 1999; published electronically Novemb er 2, 1999. http://www.siam.org/journals/simax/21-2/33837.html Institute of Mechanics, Moscow State Lomonosov University, Michurinsky pr. 1, 117192 Moscow, Russia (mailybaev@inmech.msu.su).

396


TRANSFORMATION OF MATRIX FAMILIES TO NORMAL FORMS

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complicated system of nonlinear equations containing parameters of the family and parameters of the normal form. Another approach has been suggested in [14], where the transformation functions have been found at a specific value of the parameter vector p by solving a nonlinear implicit matrix equation using secant, updating secant, or using Newton's method. Such a numerical approach, though useful for finding functions in a finite domain (functions are calculated at several points and then approximated), is not efficient in the case under consideration. This is connected with the local character of miniversal deformations, which are defined in the vicinity of a point in the parameter space and used for investigation of local properties of matrix families (singularities, bifurcations, etc.). For local analysis, information about derivatives of the transformation functions with respect to parameters is required. Computation of these derivatives by means of approximate methods of [14] is very time-consuming and leads to numerical errors (especially when the size of the matrix A is not small). In [14] also a method for computation of derivatives of the transformation functions for a case of one-parameter matrix family has been suggested. This family had a special form allowing transformation to a normal form which was simpler than a miniversal deformation. In the present paper a constructive method of finding functions describing the transformation of any given family of matrices to the normal form is suggested. The functions are found as Taylor series. Derivatives of the functions with respect to parameters, to determine coefficients of Taylor series, are found from a recurrent procedure using the derivatives of lower orders of these functions and derivatives of the family at the initial value of the vector of parameters. The recurrent procedure is convenient for numerical implementation, since it consists of elementary arithmetic operations. Proofs are based on properties of centralizers of matrices transformed to the Jordan form. Transformation of a family of matrices to the normal form has great significance for applications. It allows studying specific families (normal forms), which consist usually of sparse matrices depending on parameters in a simple way, and then obtaining results for arbitrary families of matrices. Miniversal deformations (without knowledge about transformation to them) have been used in [1, 2, 3] for classification of singularities of bifurcation diagrams, decrement diagrams, and boundaries of the stability domains in the case of two and three parameters. In [10] miniversal deformations have been used for finding tangent cones (linear approximations) to the stability domain at singular boundary points up to a diffeomorphism. All previous results were qualitative. To obtain quantitative results we need information about the transformation. In [4] specific linear combinations of first derivatives of the functions describing a change of parameters have been found and used to obtain necessary conditions for the stable perturbations of matrices. Other applications of miniversal deformations can be found in [6, 7], where a bound for a distance to a less generic matrix pencil and backward error bound in the procedure of calculation of polynomial roots are obtained. The method of finding functions of transformation, suggested in this paper, allows using miniversal deformations for quantitative study of different properties of matrices depending on parameters. Efficiency of the method for applications is illustrated by the problem of finding tangent cones (or parts of tangent cones) to the stability domain for singular boundary points. All types of singular boundary points are considered. The obtained results represent an extension of the paper [11], where, using another approach, tangent cones for singular boundary points of the stability domain of twoand three-parameter generic families of matrices have been found.


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The paper is organized as follows. In section 2 the method for determining derivatives of functions describing the transformation of a family of complex matrices to miniversal deformation is formulated and proved. Section 3 discusses the case of families of real matrices. Section 4 uses results of the previous sections to find tangent cones to the stability domain at singular boundary points. The conclusion gives a brief overview of the results obtained and discusses possibilities for their application to different problems. 2. Transformation of families of complex matrices to normal forms. For the following presentation we need definitions given by Arnold [1, 2], [3, p. 237]. Definition 2.1. A family of complex matrices is a holomorphic mapping A : 2 P - Cn , where P is the vicinity of some point p0 in the parameter space Cm and 2 Cn is the space of n â n complex matrices. Let us consider an arbitrary family of matrices A(p), p = (p1 ,...,pm )T Cm . Denote by i the eigenvalues of the matrix A0 = A(p0 ) and let n1 (i ) n2 (i ) ··· be the dimensions of the Jordan blocks belonging to i , beginning with the largest one. Let the Jordan form J of the matrix A0 be given by (2.1) where
- A0 = C0 JC0 1 ,

J = Ji =

J1

J2 .. . , Ji2 .. . ,

Ji1

and the Jordan block

J =
k i

i

1 ·

· ·· · 1 i

has dimension nk (i ). kl Denote by Xij the block of a square n â n matrix X, which is situated on the intersection of the rows corresponding to the block Jik and columns corresponding l kl to the block Jj ; see, for example, Figure 2.1. Denote elements of each block Xij by kl xij (r, s), r = 1,...,nk (i ), s = 1,...,nl (j ), where r, s are the numbers of a row kl and a column in Xij corresponding to xkl (r, s). In the case i = j we will leave only ij kl one number in the subscript, e.g., Xi , xkl (r, s). The block of the matrix X which i corresponds to the block Ji of the Jordan form J will be denoted by Xi . Definition 2.2. A versal deformation of a matrix A0 is a family of matrices A (p ), p Cd such that any family A(p), A(p0 ) = A0 in the vicinity of the point p = p0 can be represented in the form (2.2) A(p) = C (p)A ((p))C
-1

(p),


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399

(a)

(b)

Fig. 2.1. Block decomposition of a square matrix X according to the Jordan form J .

(a)

(b)

(c)

Fig. 2.2. Location of parameters in blocks of a miniversal deformation.

where p = (p), (p0 ) = 0 is a holomorphic mapping from the vicinity of the point p = p0 in the parameter space Cm into the vicinity of the origin of coordinate in the space Cd ; C (p) is a family of nonsingular matrices describing a change of basis. A versal deformation is said to be miniversal if the dimension d of the parameter space is the smal lest possible for a versal deformation [1, 2], [3, p. 237]. It is shown in [1, 2], [3, p. 238], that a miniversal deformation is determined by the Jordan form of the matrix A0 and can be chosen in the form of a sum J + B (p ), where B (p ) = Diag (B1 ,B2 ,...) is a family of block-diagonal matrices with blocks Bi (p ) corresponding to the blocks Ji of the matrix J . There are different ways of choosing the blocks Bi (p ). Three possible types are shown in Figure 2.2, where the block Bi (p ) has all zeros except at the places indicated. In cases (a) and (b), the indicated places are filled by different components of the vector p (parameters of the miniversal deformation); in case (c) each slanted segment is filled by a corresponding component of the vector p . Other types of blocks are also possible, e.g., blocks whose elements are all zero, except for one element on each slanted segment in Figure 2.2(c) which kl is an independent parameter. In all cases, each block Bi contains exactly nl (i ), if k l, and nk (i ), if k > l, different components of p (one on each slanted segment in Figure 2.2(c)). Denote these components by bkl (t), t = 1,..., min(nk (i ),nl (i )) i numbered inside each block from the top and from the right (see examples for the case of two Jordan blocks of the size 3 and 2 in Figure 2.3). Components bkl (t) i compose the vector of parameters of the miniversal deformation p of the dimension d = i (n1 (i )+3n2 (i )+ ···) [1, 2], [3, p. 237]. Definition 2.3. Define a generalized trace of a k â l matrix F denoted by tr(t) F as a sum of elements on that diagonal of F which is (min(k, l) - t) positions above the left lower element


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ALEXEI A. MAILYBAEV

(a)

(b)

(c)

Fig. 2.3. Numbering of parameters of a miniversal deformation.
min(k,l)

(2.3)

tr(t) F =
r =t

f (k - min(k, l)+ r, r - t +1).

In the case of a square matrix (k = l), Definition 2.3 coincides with the definition of a generalized trace given in [4]. In particular, a generalized trace tr(min(k,l)) F is equal to the left lower element of the matrix F, and a generalized trace tr(1) F of a square matrix F coincides with the ordinary trace tr F, which is equal to the sum of elements on the main diagonal. In the case of an n â n matrix F a generalized trace tr(t) of a block Fikl is equal to the sum of elements on that slanted segment which is (t - 1) positions below the longest one in this block; see Figure 2.2, tr(t) F
kl i n

=
r =t

fikl (nk (i ) - n + r, r - t +1),

n = min(nk (i ),nl (i )).

kl kl kl For blocks of the matrix B (p ) we have tr(t) Bi = i (t)bkl (t), where i (t) = 1, if Bi i kl is a block of the type (a) or (b), and i (t) = n - t +1, if Bi is a block of the type (c); see Figure 2.2. Denote by Dh the derivative

Dh =


1

|h|
m

ph ··· ph m 1

taken at the point p = p0 , where h = (h1 ,...,hm ), hi 0, is a vector whose components hi denote the order of the partial derivative with respect to pi (hi = 0 means that the derivative with respect to pi is not taken), |h| = h1 + ··· + hm is the order of the derivative. Transformation of the family of matrices A(p) to the normal form (miniversal deformation) (2.2) includes finding the family A (p ) corresponding to the Jordan form J of the matrix A0 , families C (p), C -1 (p), and holomorphic mapping p = (p) (functions bkl (t)(p)). To obtain C (p), C -1 (p), and bkl (t)(p), it is sufficient to i i determine derivatives Dh C, Dh C -1 , and Dh bkl (t) for all h. Then the desired functions i are found locally as Taylor series (by virtue of holomorphy). Theorem 2.4. Let A(p) be a family of matrices. Then the fol lowing expressions are valid for derivatives Dh , |h| > 0 of the functions C (p), C -1 (p), bkl (t)(p) transi forming the family of matrices A(p) to the miniversal deformation A (p ) described above (2.2).


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1 . (2.4) Dh bkl (t) = i tr(t) Fikl , kl i (t)
h2 h3

- F = C0

1

DhA-

h Ch 1

Dh1 CDh2A Dh3 C

-1

(2.5)

h1 +h2 +h3 =h h1 ,h2 ,h3 =h

+ A0

h Ch 1 h2 Dh1 CDh2 C
h1 +h2 =h h1 ,h2 =h

-1

C0 ,

C 2. (2.6)


h1 h2 h3 h

m

=
i=1

hi ! , hi !hi !hi ! 123

h Ch 1

h2

m

=
i=1

hi ! . hi !hi ! 12

Dh C = C0 X,
nk (i ) s

(2.7) xkl (r, s) = ij

r1 =r s1 =1

(-1)r1 -r (i - j )r1 -r+s

-s1 +1

r1 -r Cr1 -r+

s-s

1

kl gij (r1 ,s1 ), i = j,

k Cn =

n! , k !(n - k )! k l, r = 1, k l, r = 1,

(2.8)

xkl (r, s) = i



s

0,
kl gi (r - s + s1 - 1,s1 ),

-

s1 =s (r,s) r (r,s) r1 =r kl gi (r1 ,s - r + r1 +1), k > l, s = nl (i ),

0,

k > l, s = nl (i ),

s (r, s) = max(1,s - r +2), (2.9) 3 . (2.10) Dh C
-1 - = -C0 1

r (r, s) = min(nk (i ),r - s + nl (i ) - 1), G = Dh A - F.

h Ch 1 h2 Dh1 CDh2 C
h1 +h2 =h h2 =h

-1

.

The zero order derivatives Dh , h = 0 of functions C, C -1 , A (values of functions - at p = p0 ) are D0 C = C0 , D0 C -1 = C0 1 , D0 A = J . Theorem 2.4 allows finding


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ALEXEI A. MAILYBAEV

derivatives Dh of the order |h| > 0 of functions C, C -1 , and bkl (t), using only derivai tives of these functions of a lower order and the derivative DhA. Thus, (2.4)­(2.10) represent a recurrent procedure for determining the derivatives Dh C, Dh C -1 , and Dh bkl (t) for any h using derivatives Dh1A of the order |h1 | < |h| and h1 = h. i If all derivatives Dh C, Dh C -1 , and Dh bkl (t) of the order |h| s are determined, i the desired functions can be found approximately as Taylor series containing terms with derivatives Dh of the order |h| s C (p) =
0|h|s

Dh C

m r =1

(pr - p0r )h + o( p - p hr !
m
r

r

0

s

),

C

-1

(p) =
0|h|s

Dh C

-1

r =1 m

(pr - p0r )h + o( p - p hr !
r

0

s

),

bkl (t)(p) = i
0|h|s

Dh bkl (t) i

r =1

(pr - p0r )h + o( p - p hr !

0

s

).

Here p is the norm in Cm . Denoting these approximations by C (p), C -1 (p), and (p) we can represent the family of matrices A(p) in the form A(p) = C (p)A ((p))C -1 (p)+ o( p - p0 s ); that is, the recurrent procedure allows transforming an arbitrary given family of matrices A(p) to the normal form in the vicinity of the point p = p0 with the accuracy up to small terms of an arbitrary order. Note that Burke and Overton [4] obtained the formulae for calculation specific sums of the first order derivatives of functions bkl (t)(p) in the case of one parameter i p R. These formulae can be derived from expressions (2.4) and (2.5). 2.1. Example. As an example let us consider a one-parameter family of 3 â 3 matrices A(z ) which has at the point z = 0 eigenvalues 1 = 0, 2 = 2i with corresponding Jordan blocks of the order n1 (1 ) = 2, n1 (2 ) = 1, respectively, 2i + z -8z z2 4iz 1 - iz 2 . A(z ) = 2iz 2 2z -2z z2 The Jordan form of the matrix A0 = 0 010 J = 0 0 0 , C0 = 1 0 0 0 2i For the first iteration (h = 1, 11 f1 (1, 1) 11 F = f1 (2, 1) 11 f21 (1, 1) -4i G= 0 8
- A(0) and matrices C0 , C0 1 01 01 - 0 0 , C0 1 = 0 0 10 10

have the form 0 1. 0

Dh = d/dz ) we have

11 11 f1 (1, 2) f12 (1, 1) 4i 11 11 f1 (2, 2) f12 (2, 1) = -2 11 11 -8 f21 (1, 2) f2 (1, 1)

0 0 0

2i 0 , 1

0 -2i 4i 0 , 0 0

0 01 X = -4i 0 0 . -4i -2 0


TRANSFORMATION OF MATRIX FAMILIES TO NORMAL FORMS

403

Fig. 2.4. Block of a matrix commuting with the Jordan form J .

These matrices are used in (2.4), (2.6), and (2.10) to determine the first derivatives of the desired functions. Then these derivatives are used in the second iteration to find the derivatives Dh = d2 /dz 2 . As a result, the normal form A (p ) of the family A(z ) and, determined by Theorem 2.4, the functions p = (z ), C (z ), C -1 (z ) have the form 0 1 0 , p = (b1 (1),b1 (2),b2 (1))T , 0 A (p ) = b1 (2) b1 (1) 0 0 2i + b2 (1) b1 (1) = 4iz +9z 2 + o(z 2 ), -4iz +(2 - 2i)z 1 C (z ) = -4iz - 8z 2
2

b1 (2) = -2z + o(z 2 ), -2z - (5+1.5i)z 0 1+5iz 2
2 2

b2 (1) = z - 8z 2 + o(z 2 ), 1 z +(2+0.5i)z 2 + o(z 2 ), 0 -2z 2 + o(z 2 ). 1 - 5iz 2 2 2z +(5+1.5i)z

C

-1

-z - (2+0.5i)z -4iz 2 (z ) = 1 - 4iz 2

1 - 4iz 2 4iz +8z 2 4iz - (2 - 10i)z

2

2.2. Pro of of the theorem. Definition 2.5. The centralizer of a square matrix X is the set of al l matrices commuting with X . The notation is as fol lows: Z
X

= {Y : [X, Y ] = 0},

where [X, Y ] = XY - YX [1, 2, 3]. A centralizer ZJ of the Jordan form J consists of block-diagonal matrices Y = Diag (Y1 ,Y2 ,...) whose blocks Yi correspond to the blocks Ji and have the form shown in Figure 2.4 [1, 2, 3, 7]. In Figure 2.4 every slanted segment denotes a sequence of equal numbers, while other elements are all zeros. The centralizer ZJ is a plane in the 2 space of all matrices Cn which passes through the zero and identity matrix. Lemma 2.6. For any n â n matrix X and any i, k , l, t the fol lowing equality is valid: tr(
t)

[J, X ]kl = 0. i


404

ALEXEI A. MAILYBAEV

Proof. For any matrix Y ZJ we have (2.11) tr ([J, X ]Y ) = tr (JX Y - XJ Y ) = tr (YJ X - JY X ) = -tr ([J, Y ]X ) = 0.

Consider the matrix Y which has units on the tth slanted segment (beginning from the largest one) in the block Yilk (see Figure 2.4), while the other components are all zeros. Then, using (2.11), we obtain 0 = tr ([J, X ]Y ) = tr(t) [J, X ]kl . i Lemma 2.7. For any n â n matrix G, which satisfies the condition tr(t) Gkl = 0 i for al l i, k , l, t, the equation (2.12) [J, X ] = G

has the unique solution (2.7), (2.8), satisfying (2.13) xkl (1,s) = 0, i k l, xkl (s, nl (i )) = 0, k > l, i

for al l i, k , l, s. Proof. In the case when G is the zero matrix a set of solutions of (2.12) is the centralizer ZJ . To satisfy conditions (2.13) the matrix X ZJ must have zeros on the sth slanted segment (beginning from the smallest one for k l and from the largest kl one for k > l) of the block Xi ; see Figure 2.4. Hence, the equation [J, X ] = 0 has the unique solution X = 0 satisfying all conditions (2.13). Due to the linearity of equation (2.12) and conditions (2.13), if two matrices X1 and X2 are solutions of (2.12) and (2.13) for nonzero matrix G, then the matrix X = X1 - X2 satisfies the equations [J, X ] = 0 and (2.13). Consequently, X = X1 - X2 = 0; that is, if a solution of (2.12) and (2.13) exists, it is unique. By substituting the explicit form of J into (2.12) we obtain equations on the elements of the matrix X (2.14)
kl (i - j )xkl (r, s)+ xkl (r +1,s) - xkl (r, s - 1) = gij (r, s), ij ij ij

where, instead of all elements xkl with coordinates (r +1,s) or (r, s - 1), which don't ij kl belong to the block Xij , we take zeros. The fact that (2.7) and (2.8) is a solution to equations (2.12) and (2.13) can be verified by substitution of expressions (2.7) and (2.8) into (2.13) and (2.14). After substitution, equations (2.14) for i = j, r = nk (i ), s = nl (i ) - t +1, k l and for i = j, r = t, s = 1, k > l take the form tr(t) Gkl = 0. i They are satisfied by virtue of the conditions of the lemma. Other equations are satisfied identically. Proof of Theorem 2.4. The proof is divided into three parts according to the items of the theorem. Each part is denoted by a corresponding item number. Item 3 . Let us take the derivative Dh , |h| > 0 from both sides of the equality C (p)C -1 (p) = I , where I is the identity matrix:
h Ch 1 h2 Dh1 CDh2 C -1

= 0,

h1 +h2 =h

h Ch 1

h2

m

=
i=1

hi ! . hi !hi ! 12

After expressing C0 Dh C -1 in terms of the other summands and multiplying both - sides of the equation from the left by C0 1 , we will prove item 3 of the theorem.


TRANSFORMATION OF MATRIX FAMILIES TO NORMAL FORMS

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Item 1 . Let us take the derivative Dh from both sides of expression (2.2) DhA =
h Ch 1 h2 h3

Dh1 CDh2A Dh3 C

-1

,

h1 +h2 +h3 =h

h Ch 1

h2 h3

m

=
i=1

hi ! . h !hi !hi ! 23
i 1

- Expressing C0 DhA C0 1 in terms of other summands and then multiplying both sides -1 of the equation by C0 and C0 from left and from right, respectively, we obtain - - DhA = C0 1 DhAC0 - C0 1 h Ch 1 h2 h3

Dh1 CDh2A Dh3 C
-1

-1

C0 .

h1 +h2 +h3 =h h2 =h

Using equation (2.10), proved above for the term containing Dh C (2.15)
- DhA = [J, C0 1 Dh C ]+ F,

, we get

where the matrix F is defined in (2.5). Using Lemma 2.6 and taking into account that kl kl tr(t) (DhA kl ) = tr(t) (Dh Bi ) = i (t)Dh bkl (t) for |h| > 0, we will prove item 1 of the i i theorem. Item 2 . Rearranging (2.15), we obtain (2.16)
- [J, C0 1 Dh C ] = G,

where a matrix G is defined in (2.9). - - Note that C0 1 Dh C = Dh (C0 1 C ) is a derivative of the family of matrices CJ = -1 C0 C, CJ (p0 ) = I , which describe a change of basis transforming the family AJ (p) = - C0 1 A(p)C0 to the normal form, that is,
- - AJ (p) = C0 1 A(p)C0 = C0 1 C (p)A ((p))C -1 - (p)C0 = CJ (p)A ((p))CJ 1 (p). - At the point p = p0 the matrix AJ (p0 ) = C0 1 A0 C0 = J is in the Jordan form. In the proof of the versality theorem [1, 2], [3, pp. 238­241] it was shown that in this case the family CJ can be chosen lying on an arbitrary smooth surface of the dimension n2 - d in the space of all matrices which passes through the identity matrix and transversal at this point to the centralizer ZJ . For each such a surface the family CJ can be chosen in the unique way. In particular, the plane I +, where is the set of all matrices X satisfying conditions (2.13), has all these properties. The planes ZJ and I + are transversal, because the only matrix from ZJ which satisfies conditions (2.13) (intersection of the planes) is the identity matrix and the sum of dimensions of the planes is equal to the dimension of the space n2 . If we choose the plane I + - for determining CJ , we obtain that all derivatives Dh CJ = C0 1 Dh C = X must satisfy conditions (2.13), which can be regarded as normality conditions. Then the formulae of item 2 follow from Lemma 2.7 applied to (2.16) (conditions tr(t) Gkl = i kl tr(t) (DhAikl )-tr(t) Fikl = i (t)Dh bkl (t)-tr(t) Fikl = 0 are fulfilled by virtue of equality i (2.4)). Note that instead of normality conditions (2.13) one can take the same amount of any other linearly independent homogeneous equations such that the only matrix from ZJ which satisfies them is the zero matrix. Theorem 2.4 and Lemma 2.7 will remain valid if (2.8) is changed in accordance with the new normality conditions by kl solving (2.14) for i = j, which determines elements of the blocks Xi . The case of families of complex matrices A(p) depending on a vector of real parameters p Rm is the same as considered above except that the holomorphic dependence on parameters is changed to the smooth one. In this case, Theorem 2.4 is valid and allows finding derivatives Dh of functions C, C -1 , bkl (t) of the order |h| s, i where s is the degree of smoothness of the family A(p).


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3. Transformation of families of real matrices to normal forms. Let us 2 consider a family of real matrices A : P - Rn , P Rm smoothly depending on a vector of real parameters p Rm and determined in the vicinity of a point p = p0 . Let the matrix A0 = A(p0 ) have real eigenvalues i , i = 1,...,q and complex eigenvalues i , i , i = q +1,q +2,... (overline denotes the complex conjugate operation). Then the Jordan form of the matrix A0 is given by J = Diag (J1 ,...,Jq ,Jq+1 , Jq+1 ,Jq+2 , Jq+2 ,...), where the blocks Ji correspond to the eigenvalues i and have the form Ji = Diag (Ji1 ,Ji2 ,...); Jik are the Jordan blocks of dimensions nk (i ) beginning with the largest one. kl kl kl kl Denote by Xij A , Xij B , Xij C , Xij D blocks of an arbitrary matrix X of the dimension n â n which are situated on the intersection of rows and columns belonging to l l l l the blocks Jik and Jj , Jik and Jj , Jik and Jj , Jik and Jj , respectively. Denote elements kl kl kl of these blocks by xij A (r, s), xij B (r, s), xij C (r, s), and xklD (r, s). In the case i = j we ij kl will leave only one number in the subscript, e.g., XiA , xkl (r, s). iA Denote by Rl Y a decomplexification of a complex matrix Y [3, pp. 244­245] Rl Y = Re Y Im Y -Im Y Re Y .

Define a matrix JR = Diag (J1 ,...,Jq , Rl Jq+1 , Rl Jq+2 ,...). The matrices JR and J are connected by the relation JR = RJ R-1 , where R = Diag (I1 ,...,Iq ,Iq+1 ,Iq+2 ,...), R-1 = R = Diag (I1 ,...,Iq , Iq+1 , Iq+2 ,...). The blocks Ij have the form Ij = 1+ i 2 Ij -iIj -iIj Ij , i= -1,

where Ij is the identity matrix of the same dimension as Jj . Let us choose the matrix C0 transforming A0 to the Jordan form J (2.1) such that the matrices CR0 = C0 R-1 -1 - and CR0 = RC0 1 are real. For this purpose it is sufficient to choose the matrix C0 so that its columns, passing through the blocks corresponding to real eigenvalues Jj , j = 1,...,q , are real, and columns, passing through the blocks corresponding to complex conjugate eigenvalues Jj and Jj , j = q +1,q +2,..., are complex conjugate. This is possible due to the reality of A0 . Then each column of the matrix CR0 will be equal to the sum of the real and imaginary parts of the corresponding column of the matrix C0 . Using CR0 the matrix A0 can be transformed to the form JR
- -1 A0 = C0 JC0 1 = CR0 JR CR0 .

It is shown in [8], [3, pp. 244­245] that the real miniversal deformation AR (p ), p Rd , to which an arbitrary family of real matrices A(p), A(p0 ) = A0 can be transformed by a smooth change of basis CR (p) and a smooth change of parameters p = (p), (p0 ) = 0, can be chosen in the form AR (p ) = JR + BR (p ), BR = Diag (B1 ,...,Bq , Rl Bq+1 , Rl Bq+2 ,...). Here Bj is a block corresponding to the block Jj whose elements are all zero, except at the places indicated in Figure 2.3 (a, b, or c). In the blocks Bj , j = 1,...,q , the indicated places are filled by independent parameters bkl (t)a , and in the blocks Bj , j = q + 1,q + 2,..., by j complex numbers bkl (t) = bkl (t)a + ibkl (t)b , where bkl (t)a and bkl (t)b are independent j j j j j parameters. Numbering of parameters bkl (t)a , bkl (t)b is carried out in the same way j j - as in section 2. Let us introduce families of matrices C = CR R, C -1 = R-1 CR 1 ,


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A = J + B, and B = Diag (B1 ,...,Bq ,Bq+1 , Bq+1 ,Bq+2 , Bq sions AR = JR + BR = R(J + B )R-1 = RA R-1 , we obtain (3.1)

+2

,...). Using expres(p).

- A(p) = CR (p)AR ((p))CR 1 (p) = C (p)A ((p))C

-1

The family A (p ) has the same structure as the complex miniversal deformation in section 2. This family as well as AR (p ) is a miniversal deformation to which any family of real matrices A(p), A(p0 ) = A0 can be transformed in the vicinity of the point p = p0 , but it and the families C, C -1 contain complex elements. For derivatives Dh of the functions p = (p), C (p), and C -1 (p) transforming the family A(p) to the form A (p ) the formulae of Theorem 2.4 are valid if we assume bkl (t) = bkl (t)a for j j j q and bkl (t) = bkl (t)a + ibkl (t)b for j > q . Using the relations CR = CR-1 , j j j - CR 1 = RC -1 and Theorem 2.4, analogous formulae can be found for the derivatives - Dh CR , Dh CR 1 , Dh bkl (t)a , and Dh bkl (t)b . j j Theorem 3.1. Let A(p), p Rn be a family of real matrices. Then the fol low- ing formulae are valid for derivatives Dh , |h| > 0 of the functions CR (p), CR 1 (p), kl kl bj (t)a (p), and bj (t)b (p) transforming the family A(p) to the real miniversal deformation AR (p ) described above (3.1) 1 . Dh bkl (t)a = Re j tr(
t)

(R

-1

kl j (t)

FR R)kl jA

,
h Ch 1

Dh bkl (t)b = Im j
h2 h3

tr(

t)

(R

-1

kl j (t) 1

FR R)kl jA

,

-1 FR = CR0 DhA -

- Dh1 CR Dh2AR Dh3 CR

h1 +h2 +h3 =h h1 ,h2 ,h3 =h

+A0 2 .

h - Ch 1 h2 Dh1 CR Dh2 CR
h1 +h2 =h h1 ,h2 =h

1

CR0 .

Dh CR = CR0 RX R-1 , G = R-1 (Dh AR - FR )R. 3 .
- -1 Dh CR 1 = -CR0 h - Ch 1 h2 Dh1 CR Dh2 CR 1 .
h1 +h2 =h h2 =h

kl kl Here elements of the blocks XiA and XiD are expressed in terms of elements of the kl kl blocks GiA and GiD , respectively, by formulae (2.8), and elements of other blocks of the matrix X are expressed in terms of elements of the corresponding blocks of the matrix G by formulae (2.7). In the last case it is necessary to take into account that the complex conjugate eigenvalues i , i are considered independently (for example, kl to obtain the formulae for elements of the block Xij C , it is necessary to replace in kl kl (2.7) (i - j ) by (i - j ) and gij (r1 ,s1 ) by gij C (r1 ,s1 )). Proof. The proof is based on the connection of the real and complex miniversal deformations (3.1). First let us consider a structure of the complex miniversal


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ALEXEI A. MAILYBAEV

deformation A (p ) more thoroughly. As shown above, the matrix A has a blockdiagonal form with blocks AjA , j q and AjA , AjD , j > q corresponding to the real and complex eigenvalues j R, j q and j , j C, j > q , respectively. The blocks AjA depend on parameters bkl (t) = bkl (t)a for j q and on parameters j j bkl (t) = bkl (t)a + ibkl (t)b for j > q . The blocks AjD = AjA , j > q depend on paj j j

rameters bkl (t) = bkl (t)a - ibkl (t)b . Thus, we have a connection between the real and j j j complex miniversal deformations, as well as transformation functions in the form A =R
-1

(3.2)

AR R,

C = CR R, bkl (t)a , j bkl j (t)a + ibkl j

C

-1

=R

-1

- CR 1 ,

(3.3)

bkl (t) = j

j q, (t)b , j > q .

The formulae of theorem 2.4 are valid for derivatives of functions bkl (t)(p), C (p), j and C -1 (p). Combining these formulae with equalities (3.2) and (3.3), we can obtain - expressions for derivatives of bkl (t)a , bkl (t)b , CR , and CR 1 . j j Substituting relations (3.2) into formulae (2.5), (2.6), and (2.9), we obtain (3.4) (3.5) (3.6) F =R
-1

FR R,
-1

Dh CR = CR0 RX R G=R
-1

,

(Dh AR - FR )R.

It follows from expressions (3.3) and Theorem 2.4 that the derivatives Dh bkl (t)a j and Dh bkl (t)b can be found by taking the real and imaginary parts of the right-hand j side of relation (2.4) and using expression (3.4) for the matrix F . This proves item 1 of the theorem. Item 2 follows directly from relations (3.5), (3.6), and Theorem 2.4. The fact that the matrix Dh CR calculated by these formulae will be real can be proved in the following way. Elements of the matrix G = R-1 GR R, where GR = Dh AR - FR is a kl kl kl kl real matrix, have the properties gij A (r, s) = gij D (r, s), gij B (r, s) = gij C (r, s). It can be shown that elements of the matrix X obtained from such a matrix G have the same properties. It remains to note that then the matrix RX R-1 and, consequently, the matrix Dh CR = CR0 RX R-1 will be real. For the proof of remaining item 3 of the theorem it is sufficient to substitute the - relations C = CR R and C -1 = R-1 CR 1 into expression (2.10). For derivatives of the zero order h = 0 we have D0 AR = JR , D0 CR = CR0 , - -1 and D0 CR 1 = CR0 . The recurrent procedure described in Theorem 3.1 allows finding - h derivatives D of the real functions CR , CR 1 , , transforming the family of matrices A(p) to the real miniversal deformation AR (p ), for any h. Using these derivatives, the real family A(p) can be transformed to the normal form with the accuracy up to o( p - p0 s ), where s is the degree of smoothness of the family. 3.1. Example. Let us consider a one-parameter family of real matrices A(x) of the order 4 â 4 having at the point x = 0 the pair of complex conjugate eigenvalues


TRANSFORMATION OF MATRIX FAMILIES TO NORMAL FORMS

409

= 1 + i, = 1 - i with corresponding Jordan blocks of the order n() = 2: 2x2 1 1 - 4x2 1+ x -1+2x 1 - x 1 - 2x2 3x +2x2 . A(x) = 2x2 2x2 1 - 2x -1+ x 1+ x -x 3x 1+4x2 The normal forms A (p ) and AR (p ) of this family and the function p = (x) determined using the recurrent procedure of Theorem 3.1 (up to o(x2 )) have the form 1 1 -1 0 b(2)a 1+ b(1)a -b(2)b -1 - b(1)b , AR (p ) = 1 0 1 1 1+ b(1)a b(2)b 1+ b(1)b b(2)a 1+ i 1 b(2) 1 + i + b(1) A (p ) = 0 0 0 0 0 0 1-i b(2) 0 0 , 1 1 - i + b(1)

p = (b(1)a , b(1)b , b(2)a , b(2)b )T ,

b(1) = b(1)a + ib(1)b = -x - 2x2 + i -x -

21 2 x + o(x2 ), 8

b(2) = b(2)a + ib(2)b = -

7 x 25 2 3 + x + i x - x2 + o(x2 ). 2 8 2 4

4. Application to stability theory. As an application of miniversal deformations to the stability theory, let us consider a problem of investigation of the stability domain of a family of real n â n matrices A(p), p Rm in the vicinity of a singular boundary point. The stability domain of a family of matrices A(p) is a set of values of the vector of parameters p at which all eigenvalues of the matrix A(p) have negative real parts Re < 0. A boundary of the stability domain is characterized by matrices A(p) having pure imaginary eigenvalues Re = 0, while for all other eigenvalues Re < 0. Nonsingular points of the stability boundary are characterized by the existence of only one simple zero eigenvalue = 0 or one pair of simple complex conjugate eigenvalues = ±i on the imaginary axis. Otherwise the boundary point is singular [2], [3, pp. 255­256]. A linear approximation of the stability domain at the boundary point is described by a tangent cone introduced by Levantovskii [10]. Definition 4.1. A tangent cone to the stability domain at the boundary point is a set of direction vectors e = dp/d =0 of the curves p(), > 0, which start at this point and lie in the stability domain. A concept of the tangent cone can be illustrated by a simple example; see Figure 4.1. Here the tangent cone to the stability domain S = {p = (p1 ,p2 )T : 2p2 > p1 + p2 , 2p1 + p2 > p2 } at the origin has the form of an angle K = {e = (e1 ,e2 )T : 1 1


410

ALEXEI A. MAILYBAEV

Fig. 4.1. A tangent cone to the stability domain.

2e2 e1 , 2e1 e2 }. Thus, the tangent cone consists of all vectors directed into the stability domain or tangent to its boundary. In [10], using the miniversal deformation tangent cones have been described for some types of singular boundary points up to a linear diffeomorphism. In this section using the miniversal deformations and Theorem 2.4 expressions for tangent cones in the parameter space are found for all types of boundary points. It is shown that for determining the tangent cone we need only the Jordan form of the matrix A and its first derivatives with respect to parameters at the boundary point under consideration. Let p = p0 be a boundary point of the stability domain and let 0 = 0, j = ij = 0, j = -ij , j = 1,...,k be pure imaginary eigenvalues of the matrix A(p0 ) (i is the imaginary unit). Theorem 4.2. Assume that each pure imaginary eigenvalue 0 , j , j , j = 1,...,k of the matrix A(p0 ) has exactly one corresponding Jordan block of the dimension nj , j = 0,...,k. If the system of vectors f (l1 ), gj (l2 ), hj (l3 ) Rm , l1 = 1,...,n0 , l2 = 1,...,nj , l3 = 2,...,nj , j = 1,...,k , is linearly independent, where f (l1 ) = tr( (4.1)
l1 )

[F (1)11 ],..., tr( 0
(l2 ) l3 )

l1 )

[F (m)11 ] 0
l2 ) l3 )

T

,
T T

gj (l2 ) = Re tr

[F (1)11 ],..., Re tr( j

[F (m)11 ] j [F (m)11 ] j

, ,

hj (l3 ) = Im tr(

[F (1)11 ],..., Im tr( j
-1 0

F (r) = C

A C0 , pr

and derivatives are taken at p = p0 , then the tangent cone to the stability domain of the family A(p) at the point p = p0 has the form K (4.2) = {e Rm : (f (1),e) 0, (f (2),e) 0, (f (3),e) = ··· = (f (n0 ),e) = 0, (gj (1),e) 0, (gj (2),e) 0, (gj (3),e) = ··· = (gj (nj ),e) = 0, (hj (2),e) = ··· = (hj (nj ),e) = 0, j = 1,...,k }.
m

Here (x, y ) = x1 y 1 + ··· + xm y

is the scalar product in Rm . In the case when zero


TRANSFORMATION OF MATRIX FAMILIES TO NORMAL FORMS

411

is not an eigenvalue of the matrix A(p0 ) the expressions containing the vectors f (l) have to be excluded from (4.2). For the proof of the theorem we need the following lemmas. Lemma 4.3. The tangent cone to the stability domain of the family of polynomials xn - a1 xn-1 - ··· - an at the point a1 = ··· = an = 0 in the parameter space p = (a1 ,...,an )T Rn is equal to the set {a1 0, a2 0, a3 = ··· = an = 0}. Lemma 4.4. The tangent cone to the stability domain of the family of polynomials z n - (a1 + ib1 )z n-1 - ··· - (an + ibn ) at the point a1 = b1 = ··· = an = bn = 0 in the parameter space p = (a1 ,b1 ,...,an ,bn )T R2n is equal to the set {a1 0, b1 R, a2 0, b2 = a3 = ··· = an = bn = 0}. A polynomial is said to be stable if real parts of all its roots are negative. Lemmas 4.3 and 4.4 have been formulated and proved in [10]. j j Lemma 4.5. For any 1 0, 2 0, j = 0,...,k , there exists a curve j1 2 q1 (), q1 (0) = 0 in the space q1 = (al1 , bj2 , l2 = 1)T of al l coefficients (except l b1 ,...,bk ) of the polynomials 1 1 (4.3) xn0 - a0 xn 1 z
n
j 0

-1

- ··· - a0 0 , n j = 1,...,k ,

- (aj + ibj )z 1 1

nj -1

- ··· - (aj j + ibj j ), n n

such that al l polynomials (4.3) are stable along the curve at > 0, the vector of coefficients q2 = (b1 ,...,bk )T is related with q1 and by an arbitrary given smooth 1 1 function q2 = Q2 (q1 ,), Q2 (0, 0) = 0, and the fol lowing conditions are fulfil led:
j j daj j daj n j da2 j da3 1 = 1 , = 2 , = ··· = = 0, j = 0,...,k , d d d d

(4.4)

dbj j dbj n 2 = ··· = = 0, j = 1,...,k . d d

Proof. Let us consider polynomials of the form (4.5) (x + 2 )n (z + 2 )n
j 0

-2

0 0 x2 +(-1 + 2 )x - 2 + 2 ,

-2

j j z 2 +(-(1 + ibj ) + 2 )z - 2 + 2 . 1

These polynomials are stable at > 0 and their coefficients, obtained after performing multiplication, satisfy conditions (4.4). Expressing these coefficients through q2 = (b1 ,...,bk )T , and then substituting the obtained expressions into the relation 1 1 q2 = Q2 (q1 ,), we obtain q2 = Q2 (q2 ,). Moreover, the matrix of the derivatives Q2 / q2 is the zero matrix at = 0. Hence, by the implicit function theorem the vector q2 is a smooth function q2 = Q2 () in the vicinity of = 0. If we substitute the function q2 = (bj )T = Q2 () into (4.5), the coefficients of polynomials (4.5) will 1 form the desired curve q1 = q1 (). Proof of Theorem 4.2. Using expression (2.2) the characteristic equation of the matrix A(p) in the vicinity of the point p0 can be written in the form det(A(p) - E ) = det(C (p)A ((p))C
-1

(p) - E ) = det(A ((p)) - E ) = 0,

where A (p ) is the miniversal deformation of the matrix A(p0 ). Hence, stability of the matrix A(p) is equivalent to stability of the matrix A ((p)) (normal form) and, by virtue of the block-diagonal structure of A , it is equivalent to simultaneous stability of


412

ALEXEI A. MAILYBAEV

all its blocks. The blocks corresponding to eigenvalues of the matrix A0 with negative real parts are stable at the point p0 and, consequently, in the vicinity of this point. Thus, stability in the vicinity of p0 is characterized by stability of the blocks A0 , Aj , Aj , j = 1,...,k , corresponding to pure imaginary eigenvalues or, since the blocks Aj and Aj are stable or unstable simultaneously (because of complex conjugacy; see section 3), by stability of the blocks A0 , Aj , j = 1,...,k. Choosing the blocks A0 , Aj as the blocks of the type (a) (see Figure 2.2), and taking into account that there is only one Jordan block corresponding to each eigenvalue, their characteristic equations take the form det(A0 - E ) = (4.6) det(Aj - E ) =
n
0

- b11 (1)a 0

n0 -1

- ··· - b11 (n0 )a = 0, 0
nj -1

n

j

- (b11 (1)a + ib11 (1)b ) j j

- ···

··· - (b11 (nj )a + ib11 (nj )b ) = 0, j j = - ij , j = 1,...,k , where b11 (l1 )a , b11 (l2 )a , and b11 (l3 )b are parameters of the miniversal deformation 0 j j smoothly depending on p. By Theorem 2.4, and using expressions (4.1), we get (4.7) b11 (l1 )a = f (l1 ), 0 b11 (l2 )a = gj (l2 ), j = ,..., p1 pm b11 (l3 )b = hj (l3 ), j
T

.

Let us consider an arbitrary smooth curve p = p() with a direction e = dp/d. Coefficients of polynomials (4.6) along this curve have the form b11 (l1 )a () = (f (l1 ),e) + O(2 ), 0 (4.8) b11 (l2 )a () = (gj (l2 ),e) + O(2 ), j b11 (l2 )b () = (hj (l2 ),e) + O(2 ). j Applying Lemmas 4.3 and 4.4 to polynomials (4.6) and taking into account expressions (4.8), we obtain that conditions (4.2) are necessary for the curve p() to lie in the stability domain. Let us prove that conditions (4.2) are also sufficient. Denote by q a vector whose components are b11 (l1 )a , b11 (l2 )a , and b11 (l3 )b , and by q1 , q2 the parts of q so that 0 j j q2 = (b11 (1)b ,...,b11 (1)b )T and q1 consists of d = n0 +2n1 + ··· +2nk - k remaining 1 k components of the vector q . The vectors q, q1 , and q2 are connected with p by smooth functions q = Q(p), q1 = Q1 (p), and q2 = Q2 (p), which are parts of the mapping p = (p). The Jacobian dQ1 /dp is a d â m matrix whose rows f (l1 ), gj (l2 ), hj (l3 ), l3 = 1 form, according to the conditions of the theorem, a linear independent system. Let p1 be the vector consisting of d components of the vector p, which corresponds to a linearly independent system of columns of this Jacobian, and denote by p2 the vector consisting of the remaining components of p. Then det[dQ1 /dp1 ] = 0 at the point p = p0 , and by the implicit function theorem p1 = P1 (q1 ,p2 ) in the vicinity of the point p = p0 , q1 = 0. By substituting this relation into q2 = Q2 (p1 ,p2 ), we get q2 = Q2 (q1 ,p2 ). Let us consider an arbitrary direction e satisfying conditions (4.2) and assume that p2 = e2 , where e1 and e2 are the parts of the vector e corresponding to p1 and p2 , respectively. Then q2 = Q2 (q1 ,e2 ) = Q2 (q1 ,). By Lemma 4.5 there exists


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413

a curve q1 = q1 () such that dq1 /d = (dQ1 /dp)e, q2 = Q2 (q1 ,) and all polynomials (4.6) are stable at > 0 (conditions (4.4) of the lemma are fulfilled due to (4.2), (4.8)). Then the curve p = p(), > 0 such that p1 = P1 (q1 (),e2 ), p2 = e2 lies in the stability domain of the family A(p) and has the direction e, which was chosen as an arbitrary direction from the set (4.2). Theorem 4.2 covers a case when all pure imaginary eigenvalues of A(p0 ) are nonderogatory (have only one corresponding Jordan block). In the general case, when some pure imaginary eigenvalues of A(p0 ) have several corresponding Jordan blocks, the tangent cone to the stability domain is no longer a product of lines and half-lines as in (4.2). Investigation of this case is much more complicated. However, using an approach presented in the proof of Theorem 4.2, it is possible to find sets of directions belonging to the tangent cone for a boundary point of an arbitrary type. Theorem 4.6. Let 0 = 0, j , j , j = 1,...,k be pure imaginary eigenvalues of the matrix A(p0 ) and let each eigenvalue j , j = 0,...,k have corresponding Jordan blocks of the dimensions n1 (j ) n2 (j ) ···. Assume that the system of vectors rs f rs (l1 ), gj (l2 ), hrs (l3 ) Rm for r s and l3 = 1 at r = s is linearly independent, j where f rs (l1 ) = tr (4.9)
(l1 )

[F (1)rs ],..., tr( 0
(l2 )

l1 )

[F (m)rs ] 0
l2 )

T

,
T

rs gj (l2 ) = Re tr

[F (1)rs ],..., Re tr( j

[F (m)rs ] j [F (m)rs ] j

, ,

hrs (l3 ) = Im tr j

(l3 )

[F (1)rs ],..., Im tr( j
-1 0

l3 )

T

F (r) = C

A C0 , pr

and the derivatives are taken at p = p0 . Then the set K1 ={e R (4.10)
m

:

(f rr (1),e) 0, (f rr (2),e) 0, (f rr (3),e) = ··· = (f rr (nr (0 )),e) = 0,
rr rr rr rr (gj (1),e) 0, (gj (2),e) 0, (gj (3),e) = ··· = (gj (nr (j )),e) = 0,

(hrr (2),e) = ··· = (hrr (nr (j )),e) = 0, j j
rs (f rs (l1 ),e) = (gj (l2 ),e) = (hrs (l2 ),e) = 0, r > s, j = 1,...,k } j

belongs to the tangent cone to the stability domain at the point p0 . Analogously, if the system of vectors (4.9) for r s and l3 = 1 at r = s is linearly independent, then the set K2 ={e R
rr m

:

(f (1),e) 0, (f rr (2),e) 0, (f rr (3),e) = ··· = (f rr (nr (0 )),e) = 0, (4.11)
rr rr rr rr (gj (1),e) 0, (gj (2),e) 0, (gj (3),e) = ··· = (gj (nr (j )),e) = 0,

(hrr (2),e) = ··· = (hrr (nr (j )),e) = 0, j j
rs (f rs (l1 ),e) = (gj (l2 ),e) = hrs (l2 ),e) = 0, r < s, j = 1,...,k , } j

belongs to the tangent cone.


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ALEXEI A. MAILYBAEV

Proof. Characteristic equations of the blocks Aj in this case cannot be written explicitly as in (4.6). However, if we consider curves p = p() lying on the surface determined by the equations (4.12) brs (l1 )a (p) = brs (l2 )a (p) = brs (l2 )b (p) = 0, 0 j j r > s, j = 1,...,k ,

the blocks Aj will have block-triangular form, and their characteristic equations along these curves will take the form det(A0 - E ) = (4.13)
r nr (0 )

- brr (1)a 0
nr (j )

nr (0 )-1

- ··· - brr (nr (0 ))a = 0 0
nr (j )-1

det(Aj - E ) =

r

- (brr (1)a + ibrr (1)b ) j j = - ij ,

- ···

··· - (brr (nr (j ))a + ibrr (nr (j ))b ) = 0, j j

j = 1,...,k .

By Theorem 2.4 and using expressions (4.9), we obtain brs (l1 )a = f rs (l1 ), 0
rs brs (l2 )a = gj (l2 ), j

brs (l3 )b = hrs (l3 ). j j

rs Since the system of the vectors f rs (l1 ), gj (l2 ), hrs (l2 ), r > s is linearly independent, j by the implicit function theorem (4.12) can be solved with respect to a part of variables p1 = P1 (p2 ), where p1 and p2 are parts of the vector p. Then the coefficients of polynomials (4.13) depend only on the vector of parameters p2 , and the direction of any curve p1 = P1 (p2 ()),p2 = p2 () satisfies all the conditions standing in the last row of (4.10). For the proof of the theorem it is sufficient to find a curve p2 = p2 () having an arbitrary chosen direction e2 such that the vector e consisting of e1 = (dP1 /dp2 )e2 and e2 satisfies all conditions (4.10), and all polynomials (4.13) are stable along this curve (conditions in the last row of (4.10) are satisfied automatically due to the choice of e1 ). This problem coincides with the problem solved in the proof of Theorem 4.2. To prove that the set (4.11) belongs to the tangent cone, one should consider curves lying in the surface

brs (l1 )a (p) = brs (l2 )a (p) = brs (l2 )b (p) = 0, 0 j j

r < s,

j = 1,...,k .

Then the proof is carried out analogously. The relations determining sets (4.10) and (4.11) are sufficient conditions that a direction belongs to the tangent cone. Note that Burke and Overton [4] obtained necessary conditions for the stable perturbations of the matrix in the general case, which can be used as necessary conditions that a direction belongs to the tangent cone. In the nonderogatory case, sets (4.10) and (4.11) coincide and are equal to the tangent cone by Theorem 4.2. Theorems 4.2 and 4.6 give a constructive method for determining the tangent cone to the stability domain at the boundary point. For this purpose we need only information about the matrix A and its first derivatives with respect to parameters at the point under consideration. In the case of Theorem 4.2 for determining the tangent cone, we need to transform the matrix A0 = A(p0 ) to the Jordan form (2.1) and find the derivatives A/ pr at the point under consideration. Note that components of vectors (4.1) are sums of - products of some row of the matrix C0 1 by the matrix A/ pr and by some column - of the matrix C0 . Moreover, only the columns and rows of the matrices C0 and C0 1 ,


TRANSFORMATION OF MATRIX FAMILIES TO NORMAL FORMS

415

corresponding to the blocks of pure imaginary eigenvalues, are used. This fact can be used to reduce the amount of calculations for determining the tangent cone. It is well known that the columns of the matrix C0 corresponding to the block Aj n -1 (in the nonderogatory case) are the eigenvector and associated vectors u0 ,...,uj j j (also called generalized eigenvectors) of the eigenvalue j A0 u0 =j u0 , j j A0 u1 =j u1 + u0 , j j j . . . A0 u
nj -1 j

(4.14)

=j u

nj -1 j

+u

nj -2 i

.

- Then the rows of the matrix C0 1 corresponding to the same block are determined as n -1 0 the left eigenvector and associate vectors vj j ,...,vj , taken in reverse order, 0 0 vj A0 =j vj , 1 1 0 vj A0 =j vj + vj , . . .

(4.15) vj (4.16)
0 vj u nj -1 j

nj -1

A0 =j vj
nj -1 j

nj -1

+ vi

nj -2

,

and satisfying the normalization conditions = 1,
l vj u

= 0,

l = 1,...,nj - 1.

- Using these properties of the matrices C0 and C0 1 expressions (4.1) can be written in the form n0 -l
1

f (l 1 ) =
t=0

t v0

A u p1
2

n0 -l1 -t 0

n0 -l

1

,...,
t=0

t v0

A u pm
2

n0 -l1 -t 0

T

,
T

nj -l

(4.17)

gj (l2 ) = Re
t=0 nj -l
3

t vj

A u p1

nj -l2 -t j

nj -l

,..., Re
t=0 nj -l
3

t vj

A u pm

nj -l2 -t j

,
T

hj (l3 ) = Im
t=0

t vj

A u p1

nj -l3 -t j

,..., Im
t=0

t vj

A u pm

nj -l3 -t j

.

According to (4.17), for determining the vectors f (l1 ), gj (l2 ), and hj (l3 ) we need to know only right and left eigenvectors and associated vectors corresponding to pure imaginary eigenvalues and satisfying normalization conditions (4.16), and the derivatives A/ pr , r = 1,...,m at the point p = p0 . Analogous expressions can be obtained for vectors (4.9). For this purpose in exrs t pressions (4.17) one should substitute f rs (l1 ), gj (l2 ), hrs (l3 ), ut , vjr , n0 , nj instead j js t t of f (l1 ), gj (l2 ), hj (l3 ), uj , vj , n0 , nj , respectively, where n0 = min(nr (0 ),ns (0 )), t nj = min(nr (j ),ns (j )), and ut vjr , t = 0,...,nr (j ) - 1 are right and left eigenjr r vectors and associated vectors corresponding to the Jordan block Jj of the eigenvalue t1 t2 j . In this case the normalization conditions for the vectors ujr , vjr take the form t2 (4.16), while the normalization conditions for the vectors ut1 , vjs , r = s are jr v
nr (j )-1 ns (j )-t ujs jr

= 0,

t = 1,...,nj ,

j = 0,...,k .


416

ALEXEI A. MAILYBAEV

Note that Theorems 4.2 and 4.6 are valid in the case of families of complex matrices smoothly depending on a vector of real parameters, if the vectors gj (l1 ), hj (l2 ), rs and gj (l1 ), hrs (l2 ) are determined separately for each pure imaginary eigenvalue of j the matrix A0 (also for the zero eigenvalue and for each of the complex conjugate eigenvalues, if any) and the expressions containing the vectors f (l) and f rs (l) are excluded from (4.2), (4.10), and (4.11). If we consider generic families of matrices [1, 2, 3] the condition of linearly independence of the system of vectors in Theorem 4.2 is satisfied. It follows from the fact that, in the generic case, the codimension of a surface in the parameter space Rm (stratum), on which all the matrices A(p) have the same type of Jordan structure of pure imaginary eigenvalues of the matrix A0 , is equal to d = n0 +2n1 + ··· +2nk - k (the number of parameters of the miniversal deformation, which correspond to the blocks of pure imaginary eigenvalues minus the number of nonzero pure imaginary eigenvalues, whose magnitudes are arbitrary on the stratum). There is a field of vectors (4.1) determined on this surface (d vectors at each point). The condition that the system of these vectors is linearly dependent leads to m equations at each point of the surface, where components of one vector are expressed as a linear combination of corresponding components of the other vectors with (d - 1) arbitrary coefficients. These equations are equivalent to (m - d + 1) equations which do not contain these coefficients. But, since the dimension of the stratum is equal to m - d (less than the number of equations), in the generic case these equations have no solution on the stratum. This means that the system of vectors is linearly independent. The analogous statement is correct in the case of Theorem 4.6. Thus, Theorems 4.2 and 4.6 can be applied to all points of a boundary of the stability domain of a generic family of matrices. 5. Conclusion. In this paper the problem of transformation of families of matrices to the Arnold's and Galin's normal forms (miniversal deformations) is solved. A constructive method determining a change of basis and a change of parameters, transforming a family of real or complex matrices to the normal form in the vicinity of the initial value of the parameter vector, is suggested. Finding the miniversal deformation of a family of matrices and describing the transformation to it represent a complete procedure of transformation to the normal form (2.2). The first step in this procedure is transformation of the matrix A0 = A(p0 ) to the Jordan form (2.1) and finding the miniversal deformation A (p ) according to [1, 2, 3]. Then, using Theorems 2.4 or 3.1, the derivatives of the families C (p), C -1 (p) and of the mapping p = (p), with respect to the parameters at the point p = p0 , are determined. At this step a recurrent procedure is used. Note that the procedure consists of only explicit relations containing elementary arithmetic operations. Using the obtained derivatives, the desired functions p = (p) and C (p), C -1 (p) (describing a change of parameters and a change of basis) are found as Taylor series. Thus, the procedure of transformation of a matrix family to the normal form is complete. The spectrum of the normal form A ((p)) coincides with the spectrum of the family A(p), but A (p ) is a family of sparse matrices depending on parameters in a quite simple way. This makes it efficient to apply normal forms (if the mapping p = (p) is known) to investigation of different properties of the spectrum of the family A(p). In section 4 the stability domain of a family of matrices in the vicinity of the singular boundary point p = p0 is studied. In the cases when all pure imaginary eigenvalues of the matrix A(p0 ) are nonderogatory, tangent cones to the stability domain at the point p = p0 are found. In the case of an arbitrary Jordan structure


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of A(p0 ), sufficient conditions, which describe directions belong to the tangent cone, are found. These conditions describe subsets of the tangent cones. It turns out that tangent cones are determined by the right and left eigenvectors and associated vectors corresponding to pure imaginary eigenvalues and the first derivatives of the matrix A(p) with respect to parameters at the point p = p0 . Expressions determining the tangent cone are explicit and simple. They can be used in different stability problems studying dependence on parameters. Moreover, by means of using transformation to the normal form, it is possible to investigate local properties of the spectral radius, to solve the problems of finding eigenvalues of perturbed matrices, etc. In this connection it is convenient to investigate the collapse of nonderogatory multiple eigenvalues. In this case the perturbed eigenvalues are determined as roots of a polynomial whose coefficients are obtained from Theorem 2.4. Acknowledgments. The author thanks Alexander P. Seyranian for his attention to this work and helpful discussions, and Wolfhard Kliem and Michael L. Overton for useful comments.
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