The differential of a primary matrix function

209

THE DIFFERENTIAL OF A PRIMARY MATRIX FUNCTION

by R. F. Rinehart (*) (Monterey, U. S. A.)

w 1. THE HAUSDORFF DIFFERENTIAL

If ~ is a finite dimensional linear associative algebra with identity over

the real or complex field, with basis el . . . . . era, and f a function with domain

and range in _~, then f is said to be Hausdorff differentiable [1] if (a) the

component functions f i of f = E f ie i are differentiable functior, s of the xi in

x = F, xiei, and if (b) d f ~ Z m d f i d xjei is expressible in the form i.j=l Oxi

d r = y ki=iAidxBi, where dx = Y,j~ldxiei and the Ai, Bi are elements of _c4

depending only on xl . . . . , xm, and not on dx t , . . . , dx , , .

Condition (b) leads to a system of linear equations over the real or complex

field for the determination of the Ai and Bi (which are not unique) [6], and the

Ai and Bi, if they exist, involve the partial derivatives of the component

funcfions f i ( x j , . . . , x,,). Explicit formulations of the Ai, Bi have not been

developed. It is well known [6] that if ~ is normal simple, condition (a) is

sufficient for HausdorII differentiability, i.e. the equations implied in condition

(b) are always solvable for the Ai and Bi.

If ~ is c/~., the algebra of n X n complex matrices, and f is a primary

function [4], i.e. a function arising from the extension of a function f ( z ) of a

complex variable z to c ~ , then f ( X ) , Xs is known to be HausdorIf

differentiable in any open set of 9/~ whose matrices have eigenvalues which

(*) Supported by Army Research Office (Durham). 14 - R e n d , C i rc . M a t e m . P a l e r m o - S e r i e 11 - T o m o X V - A n n o 1 9 8 6

2 1 0 R . F . RINEHART

are points of analyticity of f ( z ) [2]. For this special algebra and this special

but important class of functions, it might be expected that more explicit

descriptions of the A~, B~ of condition (b) might be attainable. This indeed

turns out to be the case if the domain of f in c ~ is restricted to be in the

set CKc of complex matrices with distinct eigenvalues.

w 2. PRIMARY FUNCTIONS ON Q('C

The algebra ~ endowed witl~ the topology induced by any convenient

matrix norm is a metric space. The set q~c C 9E~. of matrices X with distinct

eigenvalues is an open set of the space since the eigenvalues are continuous

functions of the elements of X. It has been shown in [5] that if P - ' X P = = d g [ X i , . . . , Xn]=D, then D is a Hausdorff differentiable function of X

throughout q(r and that for any Xo~CXc there exists an open set 9~, XoE~C~Xc, such that in ~ , P can be taken to be Hausdorff differentiable. (These assertions

are not valid at X ' s with repeated eigenvalues).

Theorem 2.1. Let f ( z ) be a function of the complex variable z and let ~ c be

the open set of matrices of ~ . ~ with distinct eigenvalues which lie in the region

of analyticity of f(z) . Let XoE.L?c and let N o D X o be an open set on which

the matrix P in P - ' X P = dg[),i, . . . , ~,,] is a (Hausdorff) differentiable function

of X. Let f ( X ) be the extension to c ~ . of f(z) . Then on Q = No A .~c the

differential of f (X ) is expressible as

d f (X) ~- f" (X) dX -[- (d P) P- ~ f (X) -- f (X) (d P) P- ' - - f" (X) [(d P) P - ' X - - X (d P) P-'].

Proof. Let X ~ Q . Then X = Pdg[).l, . . . , X n ] p - i = D, and P and D are

(Hausdorff) differentiable functions of X [5]. Hence by the customary rules for

differentiating matrix functions of scalar variables,

I d X = (dP)dg[X 1 .... , ),n] P - ' ~ Pdg[d~.~ .... , d ~] P - ' - - P d g [),~ ..... X~] P-' (dP)P- '

(I) = ( d P ) P - ' X - - X ( d P ) P - ' -[-Pdg{dX~,...,dX~]P-'.

Now as is well known f ( X ) = P d g [ f ( ) . ~ ) , . . . , f ( ; ~ ) ] P - ' [3] and hence

d f (X) = (dP)dg[f(X~), ..., f(~.n)]P-' + P d g [ f ' ( ) , j ) d ~ . . . . , f'(Xn) d),n]P -~

(2) - - P dg [f(),~) . . . . , f(),,)] P - ' (d P) P - '

= (dP)P- ' f (X) - - f (X) (dP) P-'-l-Pdg [ff(),,) ..... f'(),~)]dg[dX, .... , d),n] P- ' .

T H E DIFFERENTIAL OF A PRIMARY MATI~IX FUNCTION 2 1 1

Noting that Pdg[f'(X~) . . . . , f'(~.,)] P - ' = f ' ( X ) where f ' (X ) is the c~ -ex t ens ion

of if(z), and substituting from (1) for Pdg[dXt, . . . . dXn]P -1, (2) yields,

(3) df(X) = (dP)P-~f(X)-- f (X) (dP)P -~ + f ' ( X ) [ d X - - (dP)P- tX+ X(dP)P-~],

the formula of the theorem.

It may be noted that since dg[f'Ou)d),i , . . . . f'().n)d).n] could also be

factored in the order dg[d), i , . . . , d),,], dg [f ' () ' t ) . . . . , f ' 0",)], another equivalent

expression can be obtained in which f ' ( X ) occurs as a right factor.

Since P is differentiable in Q, dP is representable in the form d P = XHidXKi. Equation (3) therefore provides a Hausdorflian expression for df(X) of the form

required in condition (b), since P depends only on X. The expression (3)

exhibits the dependency of the Ai and Bz on the function f in explicit form;

the other contributions to the coefficients depend only on X, are independent

of the function f. Further, the only inexplicitness occurs in the recurring

factor (dP)P -~ .

w 3. REAL ARGUMENT MATRICES

It has been oullined in [5] how the concepts of Hausdorff differentiability

and Hausdorff derivative can be adapted to the case of a function whose domain tz n

is in c~R, the algebra of real mafrices and whose range is in c??~c. It was

shown that in this case the matrices P such that P - ' X P ~ - d g [ ) . t , . . . , ~] could be taken to be globally defined and differentiable on the entire open set

~R of real matrices with distinct eigenvalues, by choosing the columns of P

as eigenvectors of unit Hermitian length.

If now f (z) is a function such that f ( z ) = f ( z ) and is analytic at the

eigenvalues of a real matrix X, then the primary function f (X) is real matrix

valued [4]. For such a function f(z) , the corresponding statement of theorem 2.1

for real matrices is also valid for the global domain .~R, where Z?n is the open

set of matrices of 9]~.~ whose eigenvalues are distinct and are points of analyticity

of / (z ) .

w 4. THE MATRIX ( d P ) P -1

Since, ab initio, p - i involves the eigenvalues ;~i and dP involves the X t and

their differentials, the elements of dP and p-X, expressed as functions of the

elements of X and their differentials, will involve the elements of X irrationally.

212 R.r. mr~ruARr

However, for a large class of domains the matrix P can be so chosen that

the product ( d P ) P -~ is, rather surprisingly, rational in the elements of X, as

will be shown.

Let ~ be either the real field c)~ or complex field C, let eSe denote the

field F ( x i i , x ~ , . . . , x ,u) of all rational functicns of xi~, x~2 . . . . , x,,, over

r and let c~e-----de(~.~, . . . , ~.n) denote the splitting field of det (X- - ;~ I ) .

L e m m a 4.1. Each differential d~.i, i-----1, . . . . n, is a linear homogeneous

function of the differentials d x r s , r, s = 1, . . . , n, with coefficients belonging

to ~e. P r o o f . Each of the power sums Z" m .. i=~.~, m-----1, . , n is a rational integral

function of the coefficients of ~. in dot ( X - - ) ~ I ) ~ 0 and hence is a rational

integral function of the xrs, p,,,(x~s). Taking differentials of each of these

identities yields

(4.1) ~=1 )J~-'d~ ~ dp, , , (xr~)/m, m ~ I, . . . , n.

' T h e right members of these equations are linear homcgeneous polynomials in

the dx~ , , whose coefficients belong to c~e. The matrix of coefficients of the

d~.~ in 4.1 is the Vandermonde matrix associated with ~.~, X~, . . . , ~ . It is

nonsingular in the domain eke of matrices over ~ with distinct eigcnvalues,

and its elements belong to 3e . Hence in CKF, 4.1 is uniquely solvable for the

d ~.,, and each d),,. is a linear homogeneous function of the dx~, with coefficients

belonging to ~e.

Let @h denote the set of matrices X of eke for which column h of each

of the matrices ( X - - ~ i I ) A, i = 1 . . . . , n is not zero. Since the elements of

each of the matrices ( X - - X i I ) A are continuous functicns of the elements of X,

@, is an open subset of eke. Each nonzero column of ( X - ;~l)a is an

eigenvector of X corresponding to the eigenvalue )'i. Hence the matrix P whose

i-th column P~, i = 1 . . . . , n, is column h of ( X - - ) ~ , I ) a is a nonsingular

matrix such that P - ~ X P - - - - d g ( Z ~ , . . . , ~,) and is a differentiable function

of x [5]. Let ~ be the Galois group of the equation det ( X - ) ~ I ) ~ 0, whose

coefficient field is eSe. Let toe s be any permutation of the roots ~.~, . . . , )., of

the equation. The associated set of eigenvectors P~, . . . , P,, will undergo the

same permutation. If Z represents the permutation matrix obtained by applying

the permutation r to the columns of the identity matrix, then the matrix P

T I I E D I F F E R E N T I A L OF A P R I M A R Y M A T R I X F U N C T I O N 213

above becomes Po, = P Z , d P becomes dP, , = ( d P ) Z, and P- ' becomes -i _~p-i. , ). - 1 P,o = Z Hence, under the permutation to of the roots Xt, ... ,,(dPo,)Po, - -

( d P ) Z Z - ~ P - I ~ ( d P ) P -I and ( d P ) P -~ is invariant under any to of Q. This

also follows in exactly the same way if, in the real case, the columns of P

are normalized to Hermitian length one.

The elements of P, being cofactors from the matrices X - - ~ . z l , belong to

3F. Hence the elements of P - ' also belong to c~p. The elements of d P are

differentials of elements of c~r and therefore are linear homogeneous functions

of the differentials dx,s and dX~, with coefficients in c~r. Therefore, by Lemma

4.1. the elements of d R are expressible as linear homogeneous polynomials

in the dx,s with coefficients in c~. This, together with the invariance under -Q

implies that the coefficients of the dx,s in ( d P ) P - ' belong to dr=, i.e. are

rational functions of the x,s. If in the case c3:-~-c~ each column P~ of P is normalized by dividing by

its hermitian length, the conclusions of the preceding paragraph remain valid.

For let P: denote the corresponding normalized vector, and let P' denote the

matrix of these column vectors. Then the matrix P' of normalized vectors is

P'--= P d g ( l ~ '/', . . . , I~-'/~), where lj denotes the square of the hermitian length

of Pi. Then p.-1 dg(l'J' 1/o -1 , . . . , 1~ '1 , )_ = , . . . . l,,-)P , and d P ' = ( d P ) d g ( l ~ '/'

. . . . . l-~/, d Pdg(___l_l_~/ ,d! 1 ,, / , , )and 2 -t - - 1 . . . . , 2

( 1 -t l l : l ) d g ( d l , ' , d l , ) p - l . (4.2) ( d P ' ) P ' - ' = ( d P ) p - t - - P d g - - - 2 - 1 i . . . . . 2 "'"

As already shown, the elements of ( d P ) P -~ are linear homogeneous polynomials

in the differentials dx~, with coefficients belonging to cSF. In the second term

( 1 1 1:') and of in the above relation, the elements of P, of d g - - ~ l - ~ 1, . . . , 2

p-1 are all in ~,~., since ~zEc3p implies ~ E dp. Since 11 . . . . . I,, are in c~e, the

differentials dlI, . . . , dl~ are linear homogeneous polynomials in the dx~, with

coefficients in ~F by Lemma 4.1.

The invariance of ( d P ) P -~ and ( d t Y ) P '-I under Q has already been shown.

Hence the invariance under f~ of the second term of the right member of

4.2 follows. Hence the coefficients of the dx~, in 4.2 are in cSF. Thus,

Theorem 4.1. Let "7)h be the open set of matrices of C'K~ for which column

h of each of the matrices (X - - ),,. I)A , i = 1 . . . . , n is nonzero. Then in @hP

2 1 4 R . F . RINEHART

can be so chosen that the elements of (dP)P-' lie in the module of linear

homogeneous polynomials in the differentials dxrs with coefficients which are

rational functions of the elements of X. If ~" is the real field, the columns of

P may further be taken to be vectors of unit Hermitian length.

An explicit formula for (dP)P -1 in terms of X and dX would be desirable.

For second order matrices, a direct calculation of P, P - ' and dP, rather lengthy

for inclusion here, has yielded

( o o ) (: Hi - - S , K 1 =

- - 2X~2X~t - - X i 2 ( X 2 2 - - Xii )

( o o) (: :) H~ = S , K s =

- - x ~ (x~ . - - x , , ) 2 x',,

x2t (xii - - X2~) 2 x~2 X21

(o o ) H 4 = S

where

S = {x~ [(x~ - - x2~) ~" = 4x~x2~]l-'.

The quantity (x~ l - x~)~+ 4x,~x~L is equal to ( ;~2-) . t )~, the discriminant of

the characteristic equation of X, and is hence nonzero throughout CKc, the open

set of matrices X with distinct eigenvalues. The above expression for (dP)P -2 is

therefore valid throughout the region of CKc in which x~e :~ 0.

Unfortunately neither the method of calculation nor the form of the results

for c ~ suggest a mode of extension to matrices of arbitrary order.

Monterey (California), July 1965.

THE DIFFERENTIAL OF A PRIMARY MATRIX FUNCTION 2 1 5

REFERENCES

[I] Hausdorf/ F., Zur Theorie tier Sysleme comFlexer Zahlen, Leipziger Berichte, v. 52 (IC00),

pp. 43-61.

[2] Portmann W. O., Hausdorff-analytic functions of matrices, Proc. Amer. Math. Soc., v. 11

(1960), pp. 97-101.

[3] Rinehart R. F., The equivalence of definiticns of a matrix function, Amer. Math. Monthly, v. 62, no. 6 (1955), pp. 395-414.

[4] Rinehart R. F., Elements of a tt.eory of intrinsic functicns on algebras, Duke Maih. Jour., v. 27, (1960), pp. 1-20.

[5] Rinehart R. F., P and D in P - I X P = d g [ k ~ . . . . . xn] = D as matrix function of X,

Submitted to Can. Jour. of Mathematics. [6] Ringleb F., Beltrdge zur Fanktionentheorie in hyperkomplexen Systemen, Rend. d. Circ.

Matem. di Palermo, v. 57 (1933), pp. 311-240.

[7] ZurmUhl R., Matrizen, 4th Ed., Springer-Verlag, (1964).