The algebraic theory of latent projectors in lambda matrices

The Algebraic Theory of Latent Projectors in Lambda Matrices* E. D. Denman, J. Leyva-Ramos, and G. J. Jeon Department of Electrical Engineering University of Houston Houston, Texas 77004

Transmitted by Robert Kalaba

ABSTRACT Multivariable systems and controls are often formulated in terms of differential equations, which give rise to lambda matrices of the form A(A) = AaA” + A,X”-’ + . . . + A “. The inverses of regular lambda matrices can be represented by the latent projectors or matrix residues that have very specific properties. This paper describes the general theory of latent roots, latent vectors, and latent projectors and gives the relationships to eigenvalues, eigenvectors, and eigenprojectors of the companion form.

1.

INTRODUCTION

Multivariable systems such as a finite-element model of vibrating structures, control systems, and large-scale systems can be formulated in terms of second- or higher-order matrix differential equations. Although such systems can be reformulated in state-variable form, it may be more efficient, from a numerical viewpoint, to analyze the system using the higher-order differential equations. To illustrate, assume that the dynamics of a system can be characterized by the nonhomogeneous matrix differential equation A d”x I A d”-lx + a-- +A,x=f(t),

‘dt”

where Ai E Rmxm, x(t)E

1 &n-l

R*, and f(t)E

(14

R”, from which it follows that A(s)

*The authors received partial support from NASA under a grant from Langley Research Center, NSG1603. APPLIED MATHEMATZCS AND COMPUTATZUN 9273~300(1981)

0 Elsevier North Holland, Inc., 1981 52 Vanderbilt Ave., New York, NY 10017

OO9MOO3/81/08273

273 + 28902.75

274

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

given by A(s) = A&

+ . . . + A,,

+ AIF1

0.2)

will result when the Laplace transform is taken of (1.1). If s = h, then (1.2) becomes a lambda matrix (or matrix polynomial). If A, = I, then (1.1) can be defined in state-variable form and the companion matrix A, can be written as

0

I

0

...

0

0

0

I

. . .

0

0.3)

A,=

-A, - A,_, -A,_, **-

-.A,

which has eigenvalues Xi and eigenvectors yci. A, will be a mn X mn matrix with mn mn eigenvectors. When the partial-fraction expansion of [A(A)]-’ is taken, the following general form is obtained:

where r is the number of distinct latent roots given by det A( A) = 0, mi is the multiplicity of a root, and qi is equal to the nullity of A( hi) for the latent root A,. The matrices Pi,lECmXm will be called latent projectors (or matrix residues [ 11). Similarly, the companion form A,(X) is given by IX - A,, which has eigenvalues Ai given by det[ A,(X)] = 0. The partial-fraction expansion of A;‘(X) can be written as

with Pi,l~ Cmnx”” denoted as eigenprojectors or matrix residues [2]. The purpose of this paper is to formulate the algebraic theory of lambda matrices and the relationship of latent roots, latent vectors, and latent projectors to the eigenvalues, eigenvectors, and eigenprojectors of the companion form. The chain rule for latent projectors and eigenprojectors for the repeated latent root or eigenvalues will be given.

Latent Projectors in Lambda Matrices

275

This work follows the lines of the earlier work of Lancaster [3], Lancaster and Webber [4], and Dennis, Traub, and Weber [5] on lambda matrices and matrix polynomials. The reader should refer to Zadeh and Desoer [l] and Cullen [6] for material on matrix residues and projectors, as well as to the paper by Denman and Leyva-Ramos [2] for some material on eigenprojectors. It is assumed that the reader is familiar with the fundamentals of linear algebra. An excellent source of general material on linear algebra is Gantmacher [7]. Lambda matrices are discussed by Lancaster in [3], [4], and

PI. 2.

LAMBDA MATRICES

AND COMPANION FORMS

A system of differential equations representing a physical model may be given by jq

0

d”x(t)+A 1 d”-‘x(t) + . .. +A (&n-l &”

n

+>

q(t)

(2-l)

with initial conditions x(O), x(O), . . . ,x’“-“(0)~ R”, where Ai E Rmxm, r(t)E R”, and f(t) E R’“. The existence and uniqueness of the solution to (2.1) is assumed. It is further assumed that (2.1) is Laplace transformable with the Laplace transform taking the general form

[i&s” + &s--l + f . . + A,] x(s)

= B(s),

(24

where B(s) contains the initialcondition information as well as the Laplace transform of f( t ). If s = A in (2.2), then the resulting equation can be considered as a lambda-matrix equation. The left-hand bracketed factor in (2.2) is of interest in the remaining parts of this paper. This expression in brackets, denoted A
DEFINITION2.1. Given a set of constant matrices &E Rmxm, and a scalar X such that hi C, then a lambda matrix will be defined as A
.+. +A,)=A,A(X).

(2.3)

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E. D. DENMAN, J. LEYVA-FiAMOS, AND G. J. JEON

DEFINITION 2.2. Let &A) be as defined in (2.3), and let Ai E C be a latent root of x(h), where a latent root Xi is a root of det x(A) = 0. If &A) is regular, then there will be mn latent roots of x(A), which are equal to the eigenvalues of A c.

THEOREM2.3. Let x(h) b e as defined in (2.1), and let the determinant A< A) be given by the general form

det x(h)

= Cohmn+ C,X”‘“-’ + . . . + C,, = 0.

The lambda matrix will have at most mn-nullity&,

PROOF. Let &A)

of

(2.4)

latent roots.

be represented by

(2.5)

where each a,,(X) is a scalar polynomial of order at most IT, i.e. aif

=

aifo A” + a,llAn-l

+ . - * + aijn,

(2.6)

where aiik are the coefficients of the scalar polynomial with a iik = A,(i, i). The determinant of (2.5) has been defined by Franklin [8] as detA(h)=

2

s(ilrj2....,i,)oli,(A)aziz(h)...o,i_(A).

The summation extends over ml permutations ii, is,. . . ,j, s(jl,j2,...,im)=sign

rI

1cpcq=zm

(2.7)

of 1,2,..., m with

(i,-i,>.

The coefficient

Co is obtained from the leading coefficient given in (2.7), which will be Co = ~s(j)a1~,oa2,20. i

- - a,i,,o = det x0.

(2.8)

of det x(h)

as

(2.9)

277

Latent Projectors in Z.umbda Matrices

If det &, = 0, then A< X) will have at most mn - 1 latent roots. The coefficient

C, wiIl be given as a linear combination of the determinants of all the (m - 1)X (m - 1) submatrices of A,, Cs wilI be given as a linear combination of the determinants of ah (m -2) X(m -2) submatrices of &,, and so on until A, is the zero matrix.

COROLLARY2.3.1. There will be at least nu@y &, latent roots at the origin, i.e., h = 0. In the particular case when det A,, = 0, there is at least one latent mot at X = 0.

The proof of this corollary follows directly c mn in the summation is considered.

from (2.7) when the coefficient

DEFINITION 2.4. Let A(X) be defined as in (2.3) with matrix coefficients mi. The number of primary A, E RmX” and latent roots hi of multiplicity right or left latent vectors will be qi = nuhity A(&). The right latent vectors wiIl be denoted by yin and the left by z$“, where yjj) and ~$1) satisfy the relations

A(&)~,“‘=%zx,,

AT(&)@

with yji’~ CmX1 and z(%

= Omxl,

j=1,2

)..., Qi,

(2.10) (2.11)

i=l,2,...,o,,

CmX1.

The primary right latent vector y$‘) is a subvector of the linear’ dent right eigenvector y$) of the companion matrix of (1.3) with

indepen-

(2.12)

,

where y$) satisfies the usual algebraic equation (A, - X,Z)yL{) = OmnX1. It follows that the primary right eigenvector yi{) is a function of y,!” and Xi. The ‘Such eigenvectors

will be referred

to as primary

eigenvectors

in the rest of this paper.

278

E. D. DENMAN, J. LEYVA-FUMOS, AND G. J. JEON

primary left eigenvector zI_i)has a similar form to yri) with - @;-'I + AT,X;-2 + . . . + A;_i)$)(X-“Z

+ A;X;-3

+ +. . + AT,_2)z;i)

,CQ = C,

>

(2.13)

( hiZ + A;)+) ,.i) I where ,ii) is a primary latent vector. The two forms of y:i’ and zl_i’ given in (2.12) and (2.13) hold when qi 2 i and the maximum number of primary right or left latent vectors will be m. If mi > qi, then m, - qi generalized latent vectors must be constructed to completely define the lambda matrix from its latent roots and vectors. The mi - 9i generalized latent vectors satisfy a chain rule as given by Lancaster [4j. THEOREM2.5. y(l),

&ent

Let A(A) be defined as in (2.3). A set of right latent vectors CmX1 fm a right Jordan chain associated with the >oot hi a& the i th primary latent vector. The chain rule is given by

y(2),

. . . ,

y(hi)E

1 d2A(Ai) dA(A.) _ A(h) y!” dh2 I + 1 dX yj’ l) + 3

+ (111)1

d(‘-“A(&) dhCl_lj

_ y’r 2, + * . .

~t!‘)=%x1>

~=1,2,...,hi,

(2.14)

where y!” is the i th primary right latent vector and hi is the length of the Jordan Ah&n. The vectors yik) for 1~ k G hi are the general&d right latent vectors of the j th primary latent vector. PROOF. The proof of this theorem is obtained from consideration of the chain rule for generalized eigenvectors. The chain rule is

(A

c

-

h.Z)y’!‘= I

ct

y’?’ 3 CI

(A, - A,Z)y$’ = y,!;),

(A, -

h,J)y$h

= y$+‘,

(2.15)

279

Latent Projectors in Lambda Matrices

where y::) is the jth primary eigenvector and y$) for 1~ k G hi its generalized eigenvectors associated with the eigenvahre Xi of A,. When (2.15) is expanded, the chain rule is obtained, where yik) is formed from the first m rows of y(k). ‘I?Ie generalized right latent vectors y/l) and y,!2) will be obtained from

or (2.14) in general. The chain rule can also be utilized to modify (2.12) for the relation between the generalized eigenvectors and the generalized latent vectors with

yw t

A .y!k) + y!k-” t A?y!k’ I

y(k) ct

1

;z’A ,yV-‘) t

+ y!k-2’ t

I

(2.16)

=

THEOREM2.6. Let A(h) be defined as in (2.3), and suppose a set of left latent vectors z,!‘), zi2), . . * , .zjhi)~ CmX1 fm a left Jordan chain associated with the latent root hi and the i th primary latent uector. The chain rule is given by

AT(Xi)zi”+

~T(o dh

I

L((

,

r21 d2AT(hi) dh2

- l)+

(A)!d’-lAr(hi) #)

+

&l-l

’

=

z$‘-~)+ . . .

omxl,

1=1,2

,..., h,,

(2.17)

E. D. DENMAN, J. LEWA-RAMOS, AND G. J. JEON

280

where zi’) is the i th primaryleft latent vector and h, is the length of the Jordan chain. The vectors 2:“) for 1 c k G h, are the generalized left latent vectors of the i th primary latent vector.

PROOF. The proof of this theorem follows directly from the generalized left eigenvectors z$) of the companion form; the generalized left latent vector ‘i (k) will be formed from the last m rows of z$). The generalized left eigenvector .z$) can also be defined from the latent vectors nfk) of Xi, the latent roots Xi, and the lambda matrix A(X). Utilizing (2.13) and the chain rule for z$), it follows that the left generalized eigenvectors satisfy the relation - (A;-‘I

+ AT,X;-2 + . -. + A;_&Ik) +[(n-1)Xn-2Z+(n-2)A~X~-3f

z(k) 6-I=

.*a + AT,_,]z,!~-‘)+ **a

(A;Z + A;h, + A;)zjk) +(2&Z + A;)zik-‘)+

zik-‘)

(A,Z + A;)zjk) + xik-l) Zik) (2.18) The latent vector zik) is defined only for 1 c k G h,. The number of primary latent vectors associated with the latent root Ai is qi. and each of these latent vectors could have a chain of generalized latent vectors. The structure of a Jordan block Ii E C”Qxmi could be given by

Latent Projectors in Lambda Matrices

281

such that Zy.~rh,= mi, and every Jordan chain has a well-defined number of generalized latent vectors. The value of hi is such that the chain rules for h, + 1 as given in (2.14) and (2.17) are not satisfied. An example will now be considered to illustrate the computational proce dures. Let A(X) be defined as A(“)=[;

;]A’+[

1;:;

_;:;]A+[;:;

-;:;],

(2.20)

which has latent roots X, = 1 with multiplicity m, = 1 and h, = 2 with multiplicity m2 = 3. The right latent vectors are

and the transposed left latent vectors are

2 = [

xp

x$2’

xp’

z2

(1)

1[

-5 3

=

-2 5

-1 0

-1 5’

1

The right eigenvectors can be constructed from (2.12) and (2.16) with W, the right eigenvector matrix

w,= [

Yl’)

d”

A,yp

A,yf)

1

3

Yi2’

16 0 2

+ A2yi2)

yp

-1

Yi3’

yi"'

+ X2yi3'

I

-2 [ Yd?

l-5 -1

YS)

Yd% Y$)] *

-3

The transposed left eigenvectors can be constructed from (2.13) and (2.18) with W, the left eigenvector matrix (IX,

w,=

#‘+(ZX,+A;)Z$~)

+ A;)#) @

zp’ (Zh,+AT,)zk

z&~)+(IX~+A;)Z$~)

@

=

-8 16 3 -5

2 -9 -2

-1 5

1 3 0

1 -7 -1

-

5

1

1-122;) zi2 zg) .$q. &jr’

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

282

It should be noted that the left eigenvectors and latent vectors are reversed in order with respect to the right eigenvectors and latent vectors. The reason for this reverse ordering will be discussed later. The companion matrix A, can be generated from the right eigenvector matrix W, or the left eigenvector matrix W,: A, = WnJW;r

= W,-*JWTLY

(2.21)

where 1

1

0

0

0

0

]=O i 0

0 2

2 1’ 0IO 2

0

In the next section it will be assumed for simplicity that the 9i - 1 primary latent vectors do not have any generalized latent vectors and the sith latent vector has mi - 9i generalized latent vectors, although a more general treatment will require the structure of _Zias given in (2.19). 3.

EIGENPROJECTORS

AND LATENT PROJECTORS

It was shown in the previous section that the latent vectors of a regular lambda matrix with leading coefficient A, = I are given as subvectors of the eigenvectors of the companion matrix A, constructed from A(h). The structures of A, and A(A) as given in (1.3) and (2.3) respectively will be assumed in this section. The computation of eigenprojectors and latent projections from the eigenvectors and the latent vectors will be given in this section, and their relationship to partial-fraction expansions of [A,(h)]-’ as well as [ A(h)]-1 will be given. The chain rule for the latent projectors will be derived, and an example will be presented. Zadeh and Desoer [l] have given the general partial-fraction expansion of (AZ-A,)-’

as

[A,(A)]-~=(~z-A,)-‘=

$ i=l

misi 4,) l=l)

(A-Ai)‘+l’

(3.1)

where Pi, [E CmnXmn, mi is the multiplicity of the eigenvalue Xi, and 9i is the number of primary eigenvectors. The matrices Pi,, will be called the eigenpro jectors, since they can be constructed from the eigenvectors of A,. The eigenprojectors are also the matrix residues, which can be computed from the

Latent Projectors in Lambda Matrices

283

usual formula

pi

9

m.-qi-k I

=

k=O,l

,...,mi-oq,,

(3.2)

or from the proper selection of the right and left eigenvectors, as will be seen later. The eigenprojectors (or matrix residues) satisfy the properties of the resolution of the identity and spectral decomposition:

II.

P,,,P,

o=Pi

(), i # i,

III*

Pi,OPf.Cl=omnXmn~

Iv*

Pi, 9+1=Pi, e(Ac-XiZ),

‘* VI.

‘i,m,-qiCAc

A, = i

-

xiz)

e=o,1,2

,...,mi-q,-2,

(3.3)

=“mnXmn,

hiPi,o + i: Pi, 1.

i=l

i=l

Property VI describes the spectral decomposition of A,E Rmnxmn, and the set of eigenvalues Ai is called the spectrum of A,. An additional property can be added for defining functions of a matrix, which is obtained directly from property VI:

VII.

f(A,)=

i i=l

Pi,J(Ai)+

i i=l

mi~‘iPi,~+$$-(Ai). I=1

(3.4)

Properties I-V and VII are given by Zadeh and Desoer [l].

DEFINITION3.1. Let W,, WL~ CmnXmn be formed from the set of normalized right and left eigenvectors, z&j and Z$)E CmnX1 respectively, with WRL W* = I . Let y’Ynxmi and Z~nXm~ be the rectangular matrices (or vectors ct

284

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

when mi = 1) for hi, with ymy%

y$’

=

yL$)

...

[

y(4’-‘)

y’$’

. . .

y’l;”

1,

(3.5)

where it is assumed that the first 9i vectors in (3.5) are the primary eigenvectors and the last m, - 9i are generalized eigenvectors associated with Xi. Similarly, let

where z(r) CL3z,$f),. , , , .z(?J in (3.6) are primary eigenvectors z(yi+‘) , . . . ,z$$) are tlG generalized eigenvectors for hi. C,

and the vectors

The ordering of the left eigenvectors in (3.6) must be as shown, because the ones in the Jordan blocks for Xi are located on the subdiagonal of Ji due to the transpose operation in A,. The matrices W, and W, are the right and left eigenvector matrices (or modal matrices), with

A, = WJW,-’

where W; ’ = Wz when the eigenvectors follows from (3.7) that

= W,JW,T,

(3.7)

have been properly

normalized.

(AZ-A,)=W,(XZ-J)W,-‘=w,(Xz-Z)w,T; thus (AZ-A,)-’ of J.

= W,(XZ - J)-‘W;‘.

See the Appendix

(3.6) for the structure

THEOREM 3.2. Let Pi, ,E CmnXmn be a primary eigenprojector of(3.2) a distinct, nonrepeated eigenvalue Xi of A,. The primmy eigenprojectors given by pi o =y.z’= t t where y(f) and z,$) are nmmulized A,. ”

y(?‘xW c, CI ’

right and left eigenvectors

It

for are

(3.9) respectively

of

Latent Projectorsin La&da &OOF. Then

Matrices

Let (XI - A,)-’

285

be as given in (3.8) with pi, I defined as in (3.2).

Pi,o=Alim, (X-X,)(hZ-A,)-l=Alim(h-x,)W&Z-I)-’WL’ -t i -) I

as all diagonal elements of (h - Xi )(X Z - J)-l will be zero in the limit except for the ith diagonal element, which will be one.

THEOREM 3.3. I!.&<,~E C”“x”” be the primary eigenprojector of (3.2) for a repeated eigenvalue of A, with 9i = mi. The primary eigenprq’ectors for the repeated eigenvaks are given by

where y$) and zi{) are the rwrrrulized

eigenvectors for Xi.

PROOF. The proof follows from (3.8). The diagonal elements of (A Xi)( AZ - I)- ’ will have ones only on the diagonals of the Jordan block associated with Xi in the limit, and zero elsewhere. This means that each right eigenvector pairs off with a left eigenvector, giving (3.10).

COROLLARY3.3.1. Given that @Ai)and 22) are not normulized, then the prima y eigenprojectors are given by

(3.11)

PROOF. Assume that the normalized right and left eigenvectors by af and j3’ respectively. The numerator of (3.11) then becomes

are scaled

286

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

whereas the inner product in the denominator terms are

since z$\ry$ = 1. The product ai& in the numerator and denominator then cancel, giving (3.10). It is important to notice that ~?$~ybi) = 1 because the primary eigenprojectors satisfy property II in (3.3).

THEOREM 3.4. When the eigenvalue Xi is repeated with multiplicity mi and qi < m,, then A, will have qi primary eigenvectors and m, - qi generalized eigenvectors. The primary eigenprojector fo7 hi is given by

where the first summation is over the set of qi - 1 normu1ize.d primary eigenvectors and the second summation is over the q th primary eigenvector and the set of mi - qi normulized generalized eigenvectors.

PROOF. As before, (hZ - A,)-’ can be written as in (3.8) with J. Utilizing the material as given in Appendix A, the primary eigenprojector is

Pi,a=

lim W, tm,! h-& i

---(X-Xi)m’-qI+l(Xz-J)-l qi)! ;;z:.

w;, 1 (3.13)

which, after differentiating mi - qi times and taking the limit, gives Pi 0 =YCiz,‘i.

(3.14)

Substituting for YCiand ZCi as defined in (3.5) and (3.6) gives (3.12).

COROLLARY 3.4.1. Let @ and ~2’~’be unnorma lized right and left eigenvectors for the eigenvalue hi, whichc’has multiplicity mi, and qi < mi. The

Latent Projectors in Lambda Matrices primary eigenprojector

287

forXiisgiven

(3.15)

PROOF.

The proof of this corollary follows from the proof in Corollary

3.3.1.

COROLLARY3.42. The normalization coefficients 17;’k for the summation over the q th primary eigenvector and the generalized eigenvectors is given by

This corollary follows immediately from the definition of generPROOF. alized right eigenvectors as given in (2.18) and from the generalized left eigenvectors, as follows:

Taking appropriate inner products, the following result is obtained: $2 k = z$‘ry~;’ = #%‘r(A I

- jq)mi-Q*y$)

for qi G i 4 mi and qi G k d mi such that i + k = mi + qi; then (3.16) is proven.

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

!m3

THEOREM3.5. Let y:i) and ~2) be the right and left eigenvectors forthe eigenvalue Xi of multiplicity m, and qi < mi; then there will be mi - qi generalized eigenprojectors associated with Ai and given by

where I= 1,2 ,..., mi - qi. PROOF. This theorem is proven by considering the structure of (AZ - &)-l as given in Appendix A, and by carrying out the algebraic operations of (3.2).

COROLLARY3.5.1. Given that the secondary eigenprojectors are to be constructed j%m the set of unnormulized eigenvectors, the required rwrmuliz&ion factor is $t, qi, i.e.

1= 1,2,. . . ,mi - qi.

(3.18)

PROOF. The normalization coefficients 77:~~are constants over the qth primary and generalized eigenvectors of hi; therefore, each vector product in (3.17) will be normalized by the same constant. The algebraic theory of eigenprojectors is useful in developing the algebraic theory of latent projectors (or matrix residues) of a lambda matrix. The partial-fraction expansion of [A(A)]-’ is

MW’=

,$,yi;’ _-;i,l+l) (A

(3.19)

t

where ii l is a latent projector. It is known that ti, l can be computed in the usual manner with

2i,mi-qi-k

fork=0,1,2

=

,..., mi-qi.

&$(X-

x,)“‘qi+l[A(h)]-‘}

(3.20)

289

Latent Projectors in Lambda Matrices

The development of eigenvectors, eigenprojectors, and latent vectors will now be used to develop the algebraic theory of latent projectors as defined in (3.19) and (3.20).

will be given in the upper THEOREM3.6. TFR lumbda matrix [A(h)]-’ right m X m block of the inverse of A,(X) as shown:

.

provided

. [A(X)]-’ . . .

.

. [A,@)]-l = . . .

.

. . . .

(3.21)

that A(A) is regular and A,, = I.

PROOF. The proof of this theorem follows directly from the inverse of AZ

-z

0

0

AZ

-z

A,_,

A,_,

-0.

0

-*-

0

AC@) = i,

0..

AZ_;A,

THEOREM3.7. Let e ,, denote a primuy lutent projector of A(X) for the latent root A, of m&p&y mi, and qi = m,. The p&nay latent projector is given by

t,

o

=

x

ypz{i)T,

(3.22)

i=l where y!‘J and ziiJT are normalized right and left latent vectors respectively of A(X).

PROOF. The matrix A,(X)

can be defined by

[A,(X)]-‘=W,(XZ-J)-‘W,T,

290

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J, JEON

and from the previous theorem

=I.1 f

(hi-J)_‘[ .

.

. z’]

withY=[Y,,Yz ,..., Y,]andZ=[Z,,Z, ,..., Z,].ThelatentvectorsinY,and Zi are in sequential order, since there are no generalized latent vectors. Using the definition of an eigenprojector from (3.20) and [A(X)]-‘=Y[AZ-J]-‘ZT gives the desired results when the individual vectors of Y and Z* are considered.

COROLLARY3.7.1. given by

The lambda matrix [A(X)]-’

as defined in (2.3) is

(3.23)

[A(X)]-'=Y[XZ-J]-IZT,

where Y~CmXmn and ZECmxmn are the matrices of right and left latent vectors respectively. Equation (3.23) will be designated as the canonical form forthe inverse of a lambda matrix [3].

PROOF.

The

results stated in this corollary were obtained in the previous

theorem.

THEOREM 3.8. The primaty latent projectors gi, O of A(X) root Xi of multiplicity mi and qi i mi are given by

for the

latent

(3.24) i=l

5=4i

291

Latent Projectors in Lambda Matrices where yji) and zii) are the normulized tively.

right and left latent vectors respec-

PROOF. The proof of this follows from a detailed analysis of [A(X)]-’ defined in Corollary 3.7.1.

as

THEOREM3.9. Let Pi, ,E CmnXm” be a primary eigenprojector associated with an eigenvalue A, of multiplicity m, with qi = m,. The m X m block matrices in the last m columns of Pi,, are given by (X,)i@i,, fw i= 0,1,2 )..., n - 1, where ti, O is a primary latent prq’ector.

PROOF. When the inverse of A,(X)

is taken, the m last columns will be as

follows:

.

-1 =

L%w1-

.

. . . . . . .

. . .

.

.

Lwl -l mar' A2[A(X)]-1

(3.25)

P-‘[A.(A)]-’

But the eigenprojector Pi, o for qi = mi is defined as (3.26)

Then applying the last equation to (3.25) gives

t,O ‘i’i,

which proves the theorem.

0

(3.27)

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

292

THEOREM3.10. Let Pi, ,E CmnXmn be an eigenprojector for hi of multiplicity m, with qi primary eigenvectors and m, - qi generalized eigenvectors. The m X m block matrices in the lust m columns of Pi, o are given by r

.

@i,O

.

. . pi,O=

hji, i#,

.

.

. . .

. . .

.

.

q-l&+

(

y1

o +

o +2xiti,

)

ii,

1

1+ gi, 2

q-q.;+(

n;1)q-3$i*2+.

.. (3.28)

PROOF. This theorem is proven by considering the eigenprojector Pi, 0 defined as lim X_h,

Pi,o=

1

(mi!qi),

[ w-j

j~~~,(X-Xi)m’y,+l

*c

’

* (3.29)

When this projector is applied to (3.25), then the m X m block matrices in the last columns of Pi, ,, will have the structure given in (3.28).

THEOREM3.11. Let ti, ,E CmXm be the latent projectors associated with the regular lambda matrix A(X) with A, = 1. The set of latent projectors satisj3e.s the following properties:

i.

i

lSi,O=OmXm*

i=l

II.

i

i

i=l

I=0

(f)W+i,l=%xm

for

i=l,...,

n-2,

Z=Zmi-qi,

(3.30)

Latent Projectors in Lambda Matrices

293

If all 9i = mi, then all the generalized properties simplij$ to

IV.

i

latent prq*ectors are zero and the above

j=O,...,n-2,

qL=Omxmr

i=l

v.

i A;-l~i*o=z,x,.

(3.31)

i=l

ht pi, ,,E cmnx”” be the primary eigenprojectors PROOF. of the companion form A, as given in (3.28). When the resolution of the identity Ei=r Pi, ,, = Z is applied, the m X m block matrices in the last column satisfy the properties given in (3.30). If all 9i = mi, then Pi, 0 have a simplified structure as given in (3.27). Then the properties in (3.31) can be easily obtained. Property I in (3.30) will be called the resolution of the zero matrix. The above properties have been derived when the lambda matrix has the identity as the leading matrix coefficient, but for a general regular lambda matrix the above properties also hold where properties III and V are modified to have A,’ in their right-hand sides. It has been assumed in the theorems on latent projectors that the latent vectors were normalized with WnWl= 1. This normalization is based on complete knowledge of the eigenvectors, which may not be available. It is therefore necessary to compute the normalization factors from the latent vectors. Suppose that vi*’ is as given earlier. Then for normalized eigenvectors rl/, k = z;Pry;;‘.

(3.32)

This normalization factor must be applied to each unnormalized latent vector in the latent-projector calculation. The primary eigenvectors of A, are given in (2.12) and (2.13). Therefore the normalization factor is yp

X,yji)

sy=[. . . , z,!')~[A;Z + AlAi

+ A,],

z;“‘[ A,Z + A,], .i(“‘]

ii;y,(')

A;-;y/” =

z(,,TdA(A,) 8 _ dX ,p

for

jCqi.

(3.33)

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

294

The normalization factors for the latent vectors in the chain rule are more complicated than those for the primary latent vectors. A lemma will be given for that case. LEMMA 3.12. Let JiE Cm~Xmibe the Jordan block associated with the latent root hi, and let y,!” and zii) be the latent vectors. The normalization factor is given by

(3.34)

2

.,,i&

t+s+p=j+k+qi where hi has mi - 9i generalized ~~
latent vectors

and where

9i < t G i and

PROOF. This lemma follows from Corollary 3.3.1 and the corresponding relations for the eigenvectors and latent vectors. As stated in Corollary 3.4.2, the normalization factors nyi’q’ = $-r,q’+l= . . f = +I, mgare equal. The generalized latent projectors satisfy a chain rule, which would be expected, since they are constructed from generalized latent vectors that satisfy a chain rule.

THEOREM3.13.

ciated with pro’ojectors is

Let f’i ,E Cmx” be the generalized

latent prdectors assothe latent rdot hi. The chain rule for the generalized latent

i=O,l,...,mi_9q,,

where A(X)E

(3.35)

RmXm is a regular lambda matrix.

be the eigenprojectors of the companion form PROOF. Let Pi, l E c”“x”” A,. Using Pi, l as defined in (3.2) it is easy to show that the m X m block matrices of the last m columns have the following structure:

.. I . .

Pi,l’

f. .

*. .

t,Z Ait,,+t,Z+l

(3.36)

Latent Projectors in Lambda Matrices

295

When the structure of Pi, l is used with properties IV and V in (3.8), then the results in this theorem are obtained. The results in the last theorem also hold for a regular lambda matrix where matrix leading coefficient is different than the identity matrix. The following example will be given to illustrate some of the computational aspects of the material. Let A(X) be defined as

with latent roots h, = 1 of multiplicity 1, and X2 = 2 of multiplicity 3. The latent vectors for the lambda matrix are given by

and

z = [ zp zf’

z&2’

z2 (1)

1[ III

-;

-2

-1 5

-1 0

5’

I

Since nullityA (2) = 1, there is only one primary latent projector for X2 = 2. The latent projector for Xi = 1 can be computed directly from the latent vectors ~1’) and z(,‘) and dA(l)/dh:

The second primary latent projector t2, a is given by

where ~2 ’ is computed from (3.34) with

296

E. D. DENMAN, J. LEYVA-FiAMOS, AND G. J. JEON

It can be verified that ~32’ = 172’ = TJ~3. Th e generalized latent projectors are

and

The eigenprojectors for A, are

0 -0.6 0 -0.6 -0.6

3.2

0.6

-1

3’2

lo6

_;

0’

6

6

-0.6

4.2

0.6

_“1

p2,2

=

0 1 ’ 01 I

6

8.4 1.4 2.8

1 3

1

-3 -1 -6 -2

’

it noted the X 2 upper right blocks are the latent projectors. The eigenprojectors were computed by using W, sh Fi:,.1Wi ‘, where

297

Latent Projectors in Lambda Matrices

as constructed from (3.5), and the shifting matrices are 1

0

0

0

sw,o=i 0

0 0

0 0

0’ 0

0

0

0

0

shF2,0=i 0

0 01

0

01

0

0

0

0

0

0

se,1= i 0

shF2,2=

10’

I I

0

0

0

0

1

01 1 3

0

0

0

0

0

0

0

0

()

0

0

0’

0 [0

0 0

0 0

0

1

I

The structure of the shifting matrices has also appeared in [lo]. The partial-fraction expansion for [A(X)]-’ is

L4N-‘= 5(ALl) [-;

+

[i

5(h12)2

;]+&) [; -;] z

1 +

5(h12)3

[” -.;]. 1

The chain rule for the latent projectors t2, l satisfies the equations I.

A(2)&. 2 = 0,

II.

a@)$ 111.

as given in (3.35).

AW2,o+~

+’ 2.1

21

d242’+2 7

2=. (

E. D. DENMAN, J. LEWA-RAMOS, AND G. J. JEON

298

The results_ given in this section hold only for lambda matrices that are regular, i.e., A, is nonsingular and a full set of latent vectors exists. 4.

CONCLUSION

The theory of latent projectors has been presented and their use in the computation of the inverse of A(X) and its residues has been described. The latent projectors have specific properties and are related to the eigenprojectors of the companion form A, obtained from A(X). The concept of generalized latent projectors has been developed; they are constructed from the generalized latent vectors of A(X). The chain rule for the generalized latent projectors has been developed and described. The work on eigenprojectors, latent projectors, and lambda matrices presented in this work is not exhaustive as many other questions must be resolved; for example, the case of singular & has not been addressed. Work is currently underway to resolve the problem of singular x0 as well as other questions which arose during the work presented here. APPENDIX

A

The eigenprojectors obtained from

Pi, m,--9, _ k~ Cmnxmn, as defined in (3.2), can be

k=O,l

r*s*7mi-qi>

(A4

where W, and W, are the right and left eigenvector matrices. The inverse of hZ- Jis given by

I

(AZ-

(AZ-J)-'=

J1)-'

(AZ-

Jz)-'

.. (AZ-

Jr,-l]

(‘4.2) where Ji E CnriXn~ is the Jordan block associated with the eigenvalue Xi of multiplicity mi. For a defective matrix, the Jordan block Ji can have the

299

Latent Projectors in Lambda Matrices

following structure:

----

(A.31

%--9d+l,

where 9i is the number of primary eigenvectors and m, - 9i the number of generalized eigenvectors associated with Ai. It follows that (XI - Ji)-l is given by (AZ - 4)-l=

(h-X,)-’

(A-hi)_’ (A-Xi)-’

(A-hi)-2

...

(h-Xi)---l

(A-hi)-’

(A.41 The bracketed factor in (A.1) for XI - Ji gives (h -

Ai)mi-91+yAl

-

Ji)-l=

(X-hi)m‘-q’

(X-X‘)m’-‘I’

1

300

E. D. DENMAN, J. LEYVA-RAMOS, AND G. J. JEON

It is not difficult to evaluate Pi, m,_cl,is then given by

(A.l) for k = 0, 1,. . . , mi - qi. The eigenprojector

=W,phFi,,,-p,W:,

1sl_-----___, i

pi,0zwZ3

I

l

1

1

1

i 1 1

L_____1I

(A-6)

I

W,‘=W,shF,,,W,T.

(A.7)

I 4

The shifting matrices are important in computing eigenprojectors of a matrix when the eigenvalues and eigenvectors are known along with the multiplicity of each eigenvalue and the degeneracy of each Jordan block. REFERENCES 1

L. A. Zadeh and C. A. Desoer, Linear System Theory, McGraw-Hill, New York, 1963; R. E. Krieger, Huntington,N. Y., 1976. 2 E. D. Denman and J. Leyva-Ramos, Spectral decompositionof a matrix using the generalizedsign matrix, A&. Math. Comput. 8:237-250 (1981). 3 P. Lancaster, Lambda-Matrices and Vibrating Systems, Pergamon, New York, 1966. 4 P. Lancaster and P. N. Webber, Jordan chains for lambda matrices, Linear Algebra Appl. 1563-569 (1968). 5 J. E. Dennis, Jr., J. F. Traub, and R. P. Webber, The algebraic theory of matrix polynomials, SlAM J. Numer. Anal. 13:831-845 (1976). C. G. Cullen, Matrices and Linear Transfmtions, Addison-Wesley, 1972. F. R. Gantmacher, The Theory of Matrices, Vols. I, II, Chelsea, New York, 1960. J. N. Franklin, Matrix Theory, Prentice-Hall, Englewood Cliffs, N.J., 1968. P. Lancaster, A fundamental theorem on lambda matrices with applications, Parts I, II, Linear Algebra AppZ. 18:189-222 (1977). 10 C. F. Chen and R. E. Yates, A new approach to matrix heaviside expansion, Internat. I. Control 11(3):431-448 (1970).