Some Theorems on Holomorphic Matrices J. DE GRAAF Vorgelegt yon H. G~RTLER Abstract We consider holomorphic m x m-matrices, i.e. m x m-matrices whose matrix elements are holomorphic functions on a connected open set G in the space (7, of n complex variables. After giving some physical motivation for this research we discuss the analytic behavior of the eigenvalues, the construction of a G-adherent transformation matrix and a (physically) useful commutator theory.
w1. Introduction In physics one frequently studies propagation phenomena (waves, diffusion or combinations of both) which occur in an n-dimensional Euclidean Space R,. These propagation phenomena can often be described by vector valued functions u(x, t)=column (ul (x, t), ..., urn(x, t)). Here the u~(x, t), l <=j<_m, are real or complex valued functions which depend on the time t, 0___t < oo, and the location x e R , , x = ( x l , ..., x,). As an example we mention electromagnetic phenomena in R 3 . Then
u(x, t)=col(El (x, t), E2(x, t), E3(x, t), Bl (x, t), B2(x, t), B3(x, t)) where the E1 (x, t) . . . . . B 3 (x, t) are the components of the electric and magnetic field, respectively. Often the evolution of such a propagation phenomenon satisfies a system of linear equations with linear operator coefficients:
6a(x, t)u(x, t ) = 0 (e.g. Maxwell's equations). Special cases of (1.1) are Ou Ot ~-agu = 0 , 02 u Ot2 + ~ r
(1.1)
(1.2) (1.3)
where ~r is a matrix operator acting only on the x-coordinates. If Sr t) in (1.1) happens to be a partial differential operator with constant coefficients, application
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of a Fourier transformation with respect to the spatial variables yields
[St(k) ~-~r+...+ S l ( k ) ~ + So(k) ] ~(k, t ) = 0 .
(1.4)
Here k denotes the spatial variables (kl . . . . . kn) in the "Fourier domain" and St(k ), ..., So(k ) are polynomials in kl, ..., kn with matrix coefficients. If the propagation phenomenon under consideration occurs in a homogeneous (not necessarily isotropic) medium, the operator ~ ( x , t) can be, but need not always be, a partial differential operator with constant coefficients. In the case of differentialdifference equations or integral equations the Fourier representation of (1.1) may keep the form (1.4) while the Sl(k) ..... So(k) become general analytic matrices on some neighbourhood of the real manifold R~ of the n-dimensional complex space 6:,. Finally, we remark that Fourier transformation of (1.4) with respect to the time t yields S(k, w)~(k, co)=0. (1.5) Here co denotes the frequency variable, - oo < co< oo ; S(k, co) is an analytic matrix on some neighborhood of the real manifold R~+l of the complex space C,+ 1. The preceding considerations suggest that if one wants to study propagation phenomena in n-dimensional homogeneous (not necessarily isotropic) media, one needs a theory of holomorphic matrices of n or n + 1 complex variables. Investigations of this kind for the case n = 1 have been initiated by BROER • PELETIER in [1], [2], [3] where they discuss a representation of the solution of the initial value problem, dispersion, mode decomposition, conservation laws, group velocity and stability for systems of hyperbolic wave equations in one dimension. More general propagation phenomena in one dimension have been studied in [4], [14] where the results of BROER ~ PELETIER are extended and generalized. The mathematical methods of the subsequent sections are developed in order to provide a sound mathematical basis for the generalization of the methods of [1], [2], [3], [4], [14] to propagation phenomena in a homogeneous, not necessarily isotropic, n-dimensional medium. The author believes that the present theory applies to a still wider class of problems, namely to all evolution equations which can be transformed to a evolution equation containing holomorphic matrices. In the next sections we deal with: (i) the analytic properties of eigenvalues and transformation matrices of m x m-holomorphic matrices which are defined on a connected open set of the n-dimensional complex space Cn, (ii) a commutator theory for holomorphic matrices. One of the results is that the matrix equation X (k) B (k) = Bt (k) X (k)
(1.6)
X(k) and B(k) are matrix polynomials of the real variables kl, ..., k~ while denotes the Hermitean transposition of B(k)), possesses a solution X(k) which is positive definite for real k, if and only if (where
Bt(k)
(a) the eigenvalues of B(k) are real for real k; (b) at every point of some open subset of R n the matrix by means of a "similarity transformation".
B(k)
can be diagonalized
Holomorphic Matrices
339
w2. Fields and Vector Spaces of Analytic Functions
A field (kommutativer K6rper, Oslovaja polja) IK is a collection of objects together with four operations called addition, multiplication, subtraction and division. These operations are subject to the usual laws of elementary algebra. For the rigorous definition of a field we refer to PERLIS[5], p. 17 or VAN DERW~RDEN [6], w14. The following three types of fields which we shall use frequently are sets of single-valued analytic functions defined on a connected open set G of the space (7. of n complex variables ki, 1 _
ajl-I(kY '~
i=o 5=1 j=0
(2.1)
bj [1%)m i=1
Here aj and bj are arbitrary complex numbers. N, M, ~ij, fl~j are non-negative integers. An element (2.1) of IK(G; k) will be called an integer whenever its denominator is merely a constant. b) IK(G; ~1. . . . . ~p), where ~l(k) . . . . . ~p(k) are single-valued holomorphic functions** on G, is the field consisting of the functions N
p
aj 17[ (~,)~'J j=o ,=x M p
(2.2)
Z bj 17[((,)B,j j=0
i=l
Here aj and bj are arbitrary complex numbers; N, M, ~o, fl~J are non-negative integers. We shall call IK(G; ~1. . . . . (p) the field generated by ~1 (k) ..... ~p(k). An element (2.2) of IK(G; ~1. . . . . ~p) shall be called an integer whenever its denominator is merely a constant. c) IK(G) denotes the set of analytic functions f(k) which can be written in the form
g(k)
f(k)-- h(k)'
(2.3)
where g(k) and h(k) are holomorphic functions on G. Again, if an element (2.3)e IK(G) is such that h(k) is merely a constant, it is called an integer.
Remark 1. From the definitions it is clear that the fields a) and b) are both subfields of c).
Remark 2. Taking p = n and ( i ( k ) - k i , 1
(Ikd2+-.-+lk,12)~. ** Holomorphic on G means "regular at every point of G". See e.g. GUNr~ING& RossI [71, p. 2.
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Let IK be one of the fields a), b), c). We define I1(m as the set of all columns x = c o l (~1, ..., ~m) where the ~j are elements of IK. IK,n can be regarded as an m-dimensional vector space over the field IK. On IKm we have the operations of addition and scalar multiplication, respectively: x + y = c o l ( ~ 1 . . . . . ~m)drcol(r/1 . . . . , r/,n)=c~
dr~/1, " " , ~mdrr/rn),
UX=~tCOI(~:, ..., ~m)=col(U~l . . . . . ~ m ) ;
~ , rh, ~ e l K .
A set of vectors {Xx, x2, ..., x~}c II(m is called IK,~-independent whenever from ~tl xl + " " + ~axq = 0, where ~je IK, it follows that at = ~2 . . . . . % = 0. Further we shall consider m x m-matrices with elements in IK, in other words " m x m-matrices over the f i e l d IK". The well known theorems from elementary linear algebra on independent rows and columns, rank, determinants and Cramer's rule can all be proved for matrices over an abstract field IK. For the general theory of matrices over an abstract field IK we refer to PERLIS [5] or ~ILOV [8].
w3. Polynomials and Algebraic Functions Again let IK be one of the fields a), b), c) of w On IK we consider polynomials P(2, k ) = 2 m + a m _ x ( k ) 2 m - X + . " + a l ( k ) 2 + a o ( k ) ,
(3.1)
where ;t is a complex variable and the coefficients aj (k) are integer elements of IK. Definition 1. A polynomial (3.1) will be called IK-reducible '~ whenever it can be written as a product ()l dr bl_ 1 21- 1 dr... dr bo) (2q + Q - 12~- 1 + . . . + Co),
(3.2)
where l + q = m and the b~(k) and cj(k) are elements of IK. When this is not possible we shall call (3.1) IK-irreducible. According to VAN DER WAERDEN [6] p. 84 (or R~IFEN [9], p. 162, Satz 180), a reducible polynomial can be decomposed so that the bi and c~ in (3.2) are integers of IK. In general it depends on the choice of the field whether or not a polynomial is reducible. For example the polynomial 2 2 - k 2 is reducible with respect to IK(G; kx) and irreducible with respect to IK(G; k2). Definition 2. The ( 2 m - 1) x ( 2 m - 1) determinant 1
am- 1
1
am-2
ao
" ' "
am_ 1
am_ 2
. . . . . . . . . . . . . . . . . . . . .
.
...
det
1 al
m ( m - - 1 ) a m _ l ( m - - 2 ) a m _ 2 ...
m
(m--1)am_l . . . . . . . . . . .
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ao
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... at
associated with the polynomial (3.1) is called the discriminant o f the p o l y n o m i a l * Sometimes we shall say "reducible with respect to IK".
(3.3)
Holomorphic Matrices
341
Definition 3. The set of points k e g where the discriminant of the polynomial (3.1) vanishes shall be called the discriminantal set A (P) of the polynomial. Theorem 1. The discriminantal set A (P ) obeys one of the following two alternatives:
a) A (P) is equal to G b) A (P) is nowhere dense in G. Moreover A (P) has 2n-dimensional Lebesgue measure zero, it is closed and the set G\A (P) is connected. Proof. If the determinant (3.3) does not vanish identically, the set of its zeros is a so-called "thin subset" of G. This thin subset has the properties b). See GUNlqlNO & ROSSI [7], p. 9, corr. 10 and Chapter I, Section C. (If n = 1, a thin subset consists of isolated points.) Now we consider the polynomial equation P(2, k) = 2m+ a m _ 1 (k) 2m- 1 + . . . + ao (k) = 0
(3.4)
where the functions aj(k) are all holomorphic on a connected open domain G of the space Cn. Theorem 2. When the polynomial P(2, k) in (3.4) is irreducible with respect to some
field IK in which the coefficients aj (k) are integers (e. g. the field IK(G; ao, ..., am-2) generated by the coefficients), its discriminant does not vanish identically on G. aP. Proof. The resultant (cf. [6], p. 95) of the polynomials P and - ~ - is equal to the discriminant of P. Then from the assumption that this resultant vanishes identi~P cally it follows that P and ~ - have a common factor (see VAN DER WAERDEN [6], p. 95), which contradicts the irreducibility of P(2, k). Theorem 3. Let the polynomial P(2, k) in (3.4) be irreducible with respect to some field IK in which the coefficients aj(k) are integers. Then
(i) At every point k ~ G \ A (P ) the equation (3.4) has m distinct roots As(k) .... ,2re(k). (ii) Every point k ' ~ G \ A ( P ) has a simply connected open neighborhood 6(k'), k'~6(k'), such that on 6(k') the 2i(k), 1 < j < m , are single-valued holomorphic functions of k. (iii) At each point ~ G \ A (P) every ~j(k) can be continued analytically along any curve starting at ~ and lying entirely within G\A (P). (iv) On every simply connected open subset H of G\A (P) the m roots Aj(k) are single valued holomorphic functions. (v) Possible singularities of the 2j(k) may occur only at the points of A(P). At these singular points the 2j(k) are continuous. Proof.
(i)
According to the preceding theorem the discriminant of P0-, k) does not vanish identically because P(2, k) is irreducible. Then at every point k~ G\A (P), i.e. all points at which the discriminant of P(2, k)4=0, the equation (3.4) has m distinct solutions 21(k), ..., 2m(k).
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(ii) Follows f r o m (i) with the aid of the Implicit Function Theorem. See e.g. GUNNING & ROSSI [7], p. 14, Th. 4. (iii) In a n e i g h b o r h o o d of any point k'eG\A(P) the 2j(k) can be written as a Taylor series 2j(k) =
c i l .... l.(kl-k'l) it ..... (k.-k',) z",
~
l<=j
(3.5)
It, ...,ln=O
Let Sj(k') denote the set of points k such that m a x I k i - k~l
while (3.5) converges for every k in this set, and m o r e o v e r the positive constant pj(k') is chosen as large as possible. (If p j ( k ' ) = oo, (3.5) represents an entire function, and there are no continuation p r o b l e m s at all!) The set Sj(k') is called an elementary neighborhood for the series (3.5), while pj(k') is called the radius of this elementary neighborhood. Cf. F v g s [10] w We define p ( k ' ) = min pj(k'). Of course p(k') is the radius of the polydisc i<=j<=m
S ( k ' ) = (~ Sj(k'). Further let k"eS(k'). F o r k " we repeat the above conj=l
siderations. Then applying a convergence criterium of ABEL (see FUKS [10] w 1.3, T h e o r e m 3.3) we obtain vt
~
!
p (k) = p (k)-
!
v/
m a x I k i - ki I
.
S(k')~(k"l k)~ l <=i<=n
Fig. 1
On the other hand, since S(k') is an elementary neighborhood for at least one of the series (3.5), we have
p(k / t ) =<~p ( k ) +r
t
vt
m a x ]k~-kl [
9
l <=i
F r o m these two inequalities we obtain ] p (k") - p (k') I =< m a x [ k~ - k~'[ ~.l k. . - .k
[,
l~i
which proves that p(k) depends continuously on k. N o w let F be a continuous curve which connects the points ~, ~eG\A ( P ) . * * We suppose F to be parameterised by k(~), 0 < c t < l , k ( 0 ) = ~ , k ( 1 ) = r / . Then the function p(k(oO) 9 A set of this type is called a polydisc with radius pj(k'). 9* Each two points of a connected open set in R n c a n be joined by a (polygonal) arc lying in that open set. Cf. [15], p. 108.
Holomorphic Matrices
343
depends continuously on a. Since it vanishes nowhere, it has a minimum value fl > 0. Since k(c0 is uniformly continuous, we can choose real numbers ~o
]k(~)-k(aq)l< 89
if
aq_l
l
Now the chain of polydiscs {S(k(ctq))}~= 1 enables us to make a uniquely determined analytic continuation of every 2~(k) along F. See FUKS [10], w1.6. (iv) Follows from (iii) by applying the Monodromy Theorem. See FUKS [10] w1.6 Th. 6.1. (v) Let 2' be an s-fold root of (3.4) at k = k ' e A ( P ) . Then it follows from the Preparation Theorem of Weierstrass (see FUKS [10] w1.4 Th. 4.2) that there exists a positive constant e, such that whenever 1 2 - 2 ' 1 < e and Ik - k ' l < e the equation (3.4) can be written e (2, k) = [(2 - 4') s + a 1(k - k') (2 - 2') s-1 +.-. + A s (k - k')] f2 (4, k) = 0
(3.6)
where A l ( k - k ' ) , ..., As(k-k'), 0(2, k) are holomorphic functions of their arguments; moreover Al(k') . . . . . As(k')=0 and f2().', k')4:0. From these properties it is easily seen that the solutions 4j(k) of (3.6), which are solutions of (3.4) if [ k - k ' [ is sufficiently small, tend to 4' as k ~ k'. Definition 4. A polynomial (3.1) of which the coefficients aj(k) are holomorphic on a connected open set G ~ C~ shall be called G-coherent when all of its m roots 4j(k) can be obtained by continuing one of them analytically along paths F which lie entirely within G.
Examples. The polynomial 42 - k l , is G-coherent if and only if G contains the point kl = 0. The polynomial 4 a - e kl is not G-coherent. Theorem 4.*
I. A necessary and sufficient condition for P(4, k) to be G-coherent is that it is IK( G)-irreducible. II. I f the coefficients aj(k) are integer elements of IK(C,; k), i.e. /f they are polynomials, the condition of IK((7,; k )-irreducibility is both necessary and sufficient for P(4, k) to be C,-coherent. Proof.
I. Necessity. Suppose that P(4, k) can be decomposed P(4, k)=P1 (4, k).P2(4, k) . . . . . Pq(4, k),
q>2
(3.7)
where the coefficients of the polynomials P~, 1 < i < q, are integer elements of IK(G). Take some root, say 4t(k), of P(4, k). Now 41(k) is a root of at least one of the factors in (3.7), say P1 (2, k). Then we have P, (41(k), k ) = 0 . Obviously, analytic continuation of 41 (k) yields only roots of P1 (4, k). Therefore P(4, k) cannot be G-coherent. * In common texts (see [7], [10], [11]) one finds only a local version of this theorem.
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Sufficiency. On any simply connected open subset H of G\A (P) we can write P()I, k ) = ( 2 - 2 1 (k)). (2-22 (k)) . . . . . ( 2 - 2re(k))
(3.8)
(see Theorem 3 (iv)). We suppose that P(2, k) is not G-coherent. Then we can rename the indices j of the 2j and indicate 1+ 1 integers p ( 0 ) < p ( 1 ) < . . . 2, p(0)=0, p(l)=m such that each of the 2j, p(v-1)
P(2, k) = 1-I {(2- 2p(,_ 1)+1 (k)) . . . . . ( 2 - 2p(,)(k))}.
(3.9)
v=l
We consider the first factor in (3.9): (2-21 (k)) . . . . . ( 2 - 2p(,)(k)) = 2"(1)+ q~l (k) 2p(I)- 1 + . . . + q~po)(k). Here the coefficients are the symmetric functions
1 (k) = ;~x (k) + 22 (k) +-.. + 29(1) (k), q~2(k) = 21 (k). 22 (k) + . . . + 2p(1)- 1(k). 2p(1) (k),
(3.10)
(Pp(1)(k) = 21(k)" 22 (k) . . . . . 2p(x)(k). Obviously these functions remain single-valued under analytic continuation along any closed path in G\Z(P). Therefore the functions tpl . . . . . tpp(1) are integers in IK(G). (The possible singularities at points of A(P) can be removed; see BOCHNER [11], p. 173, Th. 5.) The other factors in (3.9) can be treated in the same way. All these factors turn out to be polynomials with coefficients which are integers in IK(G). Thus if P(2, k) is not G-coherent, it is IK(G)reducible. II. Necessity. If P(2, k) is IK(C,; k)-reducible, it is IK(C~)-reducible. Thus we can refer to I. Sufficiency. If the coefficients of P(;t, k) are polynomials defined on the whole 6", the functions q~l (k), ..., q~e(1)(k) of (3.10) are entire functions on C, which do not grow faster than some finite power Ik ]M. Then with the aid of Cauchy's integral formula (see e.g. [10], w1.2 (1.28)) it can easily be proved that the Taylor series at k = 0 of r . . . . . tPe(1) contain only a finite number of terms which are different from zero.
w4. Eigenvalues and Eigenvectors of Holomorphic Matrices Definition 5. The set of points k~C, for which the components kl, ..., k, are all real numbers is called the real manifold of C, and will be denoted by R,. Definition 6. A m x m-matrix B(k) whose elements bij(k ) are complex-valued holomorphic functions on a connected open domain G _ 6", will be called a
holomorphic m x m-matrix on G.
Holomorphic Matrices
345
Definition 7. When a holomorphic m x m-matrix can be written B(k)= ~ Bl[I(ki) a'~, j=o
(4.1)
i=1
where Bo, . . . , B N are constant complex mxm-matrices and the fl~l are nonnegative integers, B(k) will be called a matrix polynomial. Defiaition 8. With every holomorphic matrix B(k) we associate a field IK(G; B). If B(k) is a matrix polynomial we take IK(G; B)=IK(G; k); cf. (2.1). In all other cases we set IK(G; B) equal to the field which is generated by the matrix elements b~j(k); eft (2.2). In both cases G is the domain on which B(k) is defined. Theorem 5. Let B(k) be a holomorphic matrix on a domain G. Then there exists a thin subset f, of G (cf. Theorem 1 and its proof) such that I. The eigenvalues 2~(k), 1 <_a <_m of B are single valued holomorphic functions on every simply connected open subset H of G\~. II. Possible singularities of the 2~(k) may occur only at the points of/7. At the points ofF, the 2~(k) are continuous. IIL I f GnRn is non-void, there exists a point ~Gc~Rn such that some simply connected open neighborhood of ~ is a subset of G\/7. Better than that: Every neighborhood of every point 17e G n R, contains such a point ~. Proof. The eigenvalues of B(k) are roots of the polynomial equation R (2, k) = det [B (k) - 21] = 0.
(4.2)
According to GANTMACHER[12], Ch. VI, R(2, k) can be written as a product of "elementary divisors" which are powers of IK(G; B)-irreducible polynomials
R~(~,k), l<_l
(4.3)
Here the numbers s(l), 1 < l
R~(2, k)=2"q)+
~
(l~(k)2 q,
q=0 P
where the (t~(k) are integer elements of IK(G;B) and ~ s ( l ) v ( l ) = m . Further 1=1
let d~(k) be the discriminant of the polynomial R;(2, k). We take 9 equal to the set of points in G where the product d I (k). d2(k) . . . . . dp(k) vanishes. This set 27, which is the union of the discriminantal sets of the polynomials Rz(2, k) is a thin subset of G; cfi the proof of Theorem 1. According to Theorem 3 the v(l) roots 2zl(k), ..., 2:~a)(k) of each Rl(2, k) are all different from one another and are holomorphic on every simply connected open subset H of G\/~. II. Follows from the application of Theorem 3 on Rl(2, k).
I.
9 T h e p o l y n o m i a l s R t, 1 ~_ l ~ p are n o t n e c e s s a r i l y d i s t i n c t . 25 Arch. Rational Mech. Anal., Vol. 44
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III. Let Sr(tl)cG be a polydisc with radius r which is centered at tleGc~Rn. Let 9, OeSr(tl)\F, be such that the imaginary parts of their components are all positive, or all negative. Such points exist because 2; is thin. According to G t m ~ N G & RossI [7], see. C, corr. 4, the set S,0/)\2; is connected. Thus there exists a path connecting 9 with 0 and lying entirely within S,(t/)\2L Since such a path necessarily crosses the real manifold at at least one point, say 4, this point ~ is an element of S, (~/)\~. But then also a sufficiently small neighborhood of ~ belongs to S,(t/)\~. Definition 9. Let B(k) be the matrix of Theorem 5. Let H be any fixed simply connected open subset of G \ ~ . Then IK(H; B, 2~1, ..., 2 J denotes the field of single valued analytic functions on H which is generated by the generating functions of IK(H;B) and the indicated eigenvalues 2~,(k), ..., 2~(k). Obviously IK(H; B) is a subfield of IK(H; B, 2~, . . . . . 2~u). If the indicated set of eigenvalues includes all eigenvalues of B(k), we shall use the shorter notation IK(H;/]). In the formulation of Theorem 5 each one of the m characteristic roots of B(k) had its own index ~. In the sequel we shall use one index j to indicate an element 2j(k) of a certain set of equal eigenvalues. However our new notation does not mean that eigenvalues with distinct index j are necessarily different from each other. Our intention will become clearer in the next theorem. Theorem 6.
(i) On H (cf. Def. 9) the characteristic polynomial R(2, k) of B(k) can be written
as a product of elementary divisors of B(k) R(2, k ) = ( 2 - ; t l (k)) p(1). ( 2 - 2 2 ( k ) ) p(2) . . . . . ( 2 - 2 ~ ) p(~) ,
(4.4)
where p ( 1 ) + ... + p (/~)= m. (ii) On H the matrix B(k) can be written in the form
B(k) = W(k) J (k) W - X(k) ,
(4.5)
where W(k) and J(k) are both matrices over the fieM IK(H;/~) while J(k) = diag(Ja (k), ..., dj(k) . . . . . J,(k)).
(4.6)
Here Jj (k) is a p (j) x p (j)-Jordan matrix with eigenvalue 2j (k). Proof. On H=G\F, each factor R~(2, k) in (4.3) can be written as a product of linear factors. See GANT~O,CI-tER [12], VI, w or PERLIS [5], sec. 8.8.
Remarks. (i) Whenever Theorem 6 holds true for some simply connected open subset H = G\27, it is not difficult to prove (by means of analytic continuation) that the theorem holds true for any simply connected open subset of G\27. (ii) Sufficient conditions for Y(k) to be a diagonal matrix are (1) There exists a point ~ G such that B(~) has m distinct eigenvalues. (2) At each point of some open subset of G the matrix B(k) has m independent eigenvectors.
Holomorphic Matrices
347
(iii) If 2,(k) can be obtained by analytical continuation of 2j(k) we have p ( i ) = pC]). In this case the "Jordan boxes" J~(k) and Jj(k) are equally large. (iv) The transformation matrix W(k) can be written
W(k)=row(Wl(k) . . . . . Wj(k) ..... Wu(k)).
(4.7)
Here Wj(k) is a matrix with m rows and p(j) columns which is associated with the "Jordan box Jj (k)". Wj (k) can be written
Wj(k) = row (trjl (k)
. . . . .
r
(k)),
(4.8)
where aj~ . . . . . trjpr are independent vectors in IKm(H;/~). (v) If J(k) is a diagonal matrix, then p ( j ) = 1, I < j < p = m , and each column of W(k) is an eigenvector of B(k). If this is the case we shall often write
W(k) = row(w 1(k), ..., win(k)),
(4.9)
where wj(k), 1
matrix. Example. Consider the matrix
-iz
on the complex z-plane C~. On any
simply connected open neighborhood of the real axis, not including +i, this matrix is equivalent to the diagonal matrix ( V 1 07 formation matrices are
or
0 -V i ~
)
" Possible trans-
.
The second is Ct-adherent; the first is not C~-adherent. For the paths F~_,2, F2_,1 of Def. 10 one can take a path around + i (or - i ) . Theorem 7. On every simply connected open subset H of G\E the holomorphie
matrix B(k) can be brought into the Jordan normal form by means of a G-adherent transformation matrix W(k). If one wishes, W(k) can be chosen such that its elements are all integer elements of the field K(H;/~). Proof.
I.
25 *
First we consider the special case that B(k) has only one elementary divisor with respect to IK(G; B). Suppose that this elementary divisor can be written as the s th power of a IK(G; B)-irreducible polynomial Rx(;t , k) which is of degree v. Thus R (2, k) = det [B (k) - 21] = (R~ (2, k))*.
348
J. DE GRAAF:
(a) On every simply connected open domain HcG\F, the polynomial R1 (2, k) has v distinct roots 21 (k), ..., 2v(k) which are single valued and holomorphic on H. These roots are all s-fold characteristic roots of B(k) and obviously s v = m. Thus on H = G \ Z the matrix B(k) is equivalent to a generalised diagonal matrix diag (J1 . . . . . Jr), where the matrices ,/1 . . . . . Jv are all s x s-Jordan boxes with eigenvalues 21 (k), ..., 2,(k), respectively. See [12] Ch. VI w According to the preceding theorem there exists a transformation matrix W(k) over the field IK(H;/]) such that
B (k) = W(k) diag (J1 (k) . . . . . J~(k)) W-1 (k). (4.10) (b) The matrix W(k) in (4.10) can be split up according to (4.7); then W~ (k) obeys the equation
B(k) WI (k) = W~(k),l~ (k).
(4.11)
Expressed in terms of the column vectors all(k), ..., als(k ) of l,Vl(k), (4.11) becomes ( B - 2 1 I ) crll = 0 ,
(B--21I) 112=111'
(4.12)
( B - 211) crls = o'x(~- 1) 9 Now we choose a new eigenvector cr] 1 in such a way that its components are elements of IK(H; B, 21); cf Def. 9. This can always be done sine, the matrix elements of [B(k)-21(k)I ] are all elements of IK(H; B, 21). The rank of [B(k)-2t(k)I] is equal to m - l ; therefore there exists a scalar 6elK(H; B) such that a~l=6crtl. Multiplication of the second and further equations of (4.12) by 6 yields
(B - 21I) cr'12=a'11, (4.13)
-a'x(~_ x) + ( B - 21I)a'x~=O , where a [ j = r a l y , 2~j
Holomorphic Matrices
349
(d) When there exists an eigenvalue ).~(k) which we cannot reach by means
of analytic continuation of 21 (k) along some path within G\~;, we repeat the procedure (b)--(c), this time starting off with W'z(k). And so on until we have constructed m columns for the new transformation matrix
W'(k). (e) For every i, 1 < i < v, the set all, ..., a~v is a base for the invariant subspacr of IKm(H; B) which is associated with the minimal polynomial (2-2i(k)) ~. See GANTMACHER[12] Ch.VII. Then the set of all columns a~j, 1 ~i<_s, l_~j~v, establishes a base for IKm(H;/~). Therefore det[W'(k)]~0, and W' (k) can be used as a transformation matrix. II. In the general case R(2, k) can be written in the form (4.3). According to GANTMACHER[12], Ch.VI, w5 there exists a transformation matrix P(k) over the field IK(G; B) such that B (k) = P (k) O (k) P - 1(k). Here O (k) is a generalized diagonal matrix over the field IK(G; B). It has the form O (k) = diag [O 1(k), ..., Op (k)], where Ol(k ) is the v(l)s(1)x v(l)s(l)-companion matrix (Begleitmatrix) of the polynomial (Rz(2, k)) s~ See GANTMACHER[12], Ch. VI, w According to Part I of the proof, on every simply connected open set H c G \ Z we can construct a G-adherent transformation matrix Q~(k) which brings Or(k) into the Jordan normal form. On H~G\E, we introduce the generalised m x mdiagonal matrix Q(k) =diag [Q1 (k), ..., Qp(k)]. Now it is easily seen that the matrix W(k)=P(k)Q(k) is a G-adherent transformation matrix which transforms B(k) into the canonical Jordan form. III. We write our G-adherent transformation matrix W(k) in the form (4.7). If we replace each matrix W~(k) by the matrix gj (k) Wj (k), where gj (k) is the smallest common multiple of the denominators of the matrix elements of Wj(k), then the elements of the matrix gj(k)W~(k) are all integers of IK(H; B, 2j). It is not difficult to verify that the new matrix is again a G-adherent transformation matrix. w
Commutator Theory
Definition 11. The componentwise complex conjugate of an element keCn will be denoted by k. Thus k = ( k l , -.., kn). A connected open set G c Cn will be called symmetrical with respect to the real manifold Rn (or simply symmetrical) if the complex conjugate k of any element k in G is again an element of G.* Definition 12. Let f(k) be a single-valued holomorphic function on the set G of Def. 11. We define the analytic conjugatef*(k) o f f ( k ) by
f*(k)=f(~). * In many practical cases G is a symmetricalneighborhood of the whole Rn or even equal to the entire Cn.
350
J. DE GRAAF:
Theorem 8. (i) The analytic conjugate f*(k) o f f ( k ) is holomorphic on G whenever f(k) is
holomorphic on G. (ii) l f f(k) is real on some open subset of Rnc~G, then f ( k ) = f * ( k ) everywhere on G. Proof. (i) With the aid of the Cauchy-Riemann conditions (see FUKS [10], w one easily verifies that f*(k) is holomorphic on G whenever f(k) is holomorphic on G. (ii) Sincef(k)=f*(k) on some real environment of a point ~Rnc~G, we have f(k) = f * (k) on some simply connected open neighborhood of 4. See BOCHNER [1 I], p. 34, Th. 4. Then with the aid of analytic continuation it can be shown that f(k) = f * (k) everywhere on G.
Notations. Let O(k) be a vector or matrix whose elements are holomorphic on a symmetrical set G. O*(k) denotes the elementwise analytically conjugate matrix or vector. 0t(k) denotes the analytic transpose. Ot(k)=[O*(k)] r, where T denotes the transpose obtained by exchange of rows with columns. For real k, 0t(k) equals the ordinary Hermitean transpose of O(k). Remark. The analytical transpose of the matrix polynomial (4.1) can be found by replacing in (4.1) the constant matrices Bj by their Hermitean transpose B~. Definition 13. With every pair of matrices B(k), Bt(k) which are defined on a symmetrical connected open set G in Cn, we associate a field IK(G; B, Bt). If B(k) is a matrix polynomial we take IK(G; B, B t ) = IK(G; k), cf. (2.1). In all other cases we set IK(G; B, B t) equal to the field which is generated by the matrix elements bu(k ) and their analytical conjugates b*i(k). Further, if H is some connected open subset of G on which the eigenvalues of B(k) and Bt(k) are all single valued holomorphic functions, IK(H,/~,/~t) denotes the field generated by the generating functions of IK(H; B, B t) and the eigenvalues of both B(k) and Bt(k). On a symmetrical connected open set G we consider the equation X(k) B(k) = B t (k) X(k)
(5.1)
and put the question: Are there m x m-matrices X(k) over thefieM IK(G; 8, B t) which satisfy (5.1) ? The set of all m x m-matrices over a field IK can be regarded as an mZ-dimensional vectorspace IKm2.In IKm2we have the operations of addition and scalar multiplication and the notion of independent vectors. See the end of w Therefore, since (5.1) is a set of linear homogeneous equations, it makes sense to ask for the number of IKm2(G;B, Bt)-independent solutions of (5.1), or in other words, to ask for the dimension of the commutator space. Theorem 9 (the dimension of commutator space).
I.
There exists a thin subset X of G such that on every simply connected open subset H ~ G\X the eigenvalues ~1 (k) ..... Am(k) and 2~ (k), ..., 2~,(k) of B(k) and B t (k), respectively, are all single-valued holomorphic functions on H.
Holomorphic Matrices
351
lI. I f the simply connected open set H c G \ Z is symmetrical, then on H the eigenvalues of Bt (k) are the analytic conjugates of the eigenvalues orB(k). III. Let H be the set of II. Let the characteristic polynomial of B(k) be written in the form (4.4). Then the number of IKm2(G; B, Bt)-independent solutions of (5.1) is equal to ~, ~,
N= ~ y. ~(i,]).
(5.2)
i=1 j=l
Here 6(i,j) is the degree of the greatest common divisor of (2-2,(k)) p(O and (~-2*(k)) p(j). (5(i,j)>O if and only/f2,(k)=2*(k) on H.) IV. If, in addition, B(k) is similar to a diagonal matrix on H, (5.2) becomes
N= ~ ~(j),
(5.3)
j=l
where e(j) is the number of times that 2*(k) equals one of the eigenvalues 2, (k) of B(k), so 2* (k) = 2,, (k) . . . . . 2,.,~, (k). Proof.
I.
Corresponding to B(k) or Bt(k) we can construct the respective thin subsets Z 1 and E a in the manner of Theorem 5. We take Z=EI uE2, which is again a thin subset of G.
II. On any simply connected open set H c G \ Z
we can write (cf. Theorem 6)
B(k) = W(k) J (k) W -1 (k) ,
(5.4)
where W and J are matrices over IK(H; B,/~t) and J(k) is given by (4.6). For the Jj(k) we choose upper Jordan matrices. Further, if H is symmetrical, e.g. if H is a polydisc centered at ~e(G\Z)c~R,, we can take the analytic transpose of (5.4) B t (k) = W - ~ *(k) J(k) W t (k). (5.5) Now j t (k) consists of lower Jordan matrices. Obviously, the eigenvalues of j t ( k ) , which are at the same time the eigenvalues of Bt(k), are the analytic conjugates of the eigenvalues of J(k). III. On H we consider the equation
Y(k) J(k) = j t ( k ) Y(k),
(5.6)
where Y(k) is a m • m-matrix over the field IK(H;/3, ~t). Applying GANTMACHER[12], Ch.VII, w1, we find that (5.6) has N IK,~2(H;/~, Bt)-independent solutions Y(k); the number N is given by (5.2). Further, on H the equation (5.1) can be written wtxwJ = jt wtxw. (5.7) Comparison of (5.7) and (5.6) shows that on H the equation (5.1) has N IK~2(H;/~,/~t)-independent solutions X(k) which are toxin-matrices over IK(H;/~, ~t). If we collect the columns of X in one column ~', (5.1) can be rewritten F(B; B t) X = 0 (5.8)
352
J. DE GRAAr:
where F(B; B t) is a m 2 x m2-matrix over IK(H; B, Bt). The rank of F(B; B*) Therefore on H t h e equation (5.8) or (5.1) has N IKm2(H; B, Bt)independent solutions which are matrices over IK(H; B, Bt). These solutions on H have a unique analytic continuation to matrices over IK(G; B, B t) because the functions in IK(H; B, B t) are merely restrictions on the functions in IK(G; B, B*). Finally, as X(k)B(k)-B*(k)X(k)=O on the open set H, it vanishes everywhere in G. IV. Is only a special case of III. e q u a l s m 2 --N.
In the next theorem we give a relation between the eigenvectors of B(k) and the solutions of (5.1). Theorem 10. Consider a symmetrical simply connected open set H c G\7,. Suppose that on H B(k) is similar to a diagonal matrix. Let X(k) be a solution of (5.1), and suppose that for some index j, l ~_j
X(k) wj(k)=O. Proof. On H we have
X(k) [ B ( k ) - 2 j ( k ) I ] w j ( k ) = 0 . With the aid of (5.1) this becomes
[at ( k ) - 2~(k)] X (k) wj(k) = 0 . Since 8 (j) = 0, 2* (k) is not an eigenvalue of B(k). Thus ;lj (k) cannot be an eigenvalue of B t (k). This implies X(k) wj (k) = O. Theorem 11. I f the characteristic polynomial det [ B ( k ) - 2I] of B(k) is G-coherent,
we have one of the following alternatives: (a) The only solution of(5.1) is X = 0 ; (b) (5.1) has m IKm,(G; B, Bt)-independent solutions. I f (b) is the case, then on any symmetrical simply connected open set H c G \ ~ the analytic conjugate of any eigenvalue of B(k) is again an eigenvalue of B(k). ProoL Since the characteristic polynomial is G-coherent, B(k) has m distinct eigenvalues on H = G \ S . See Theorems 3 and 4. Therefore B(k) is similar to a diagonal matrix. We shall make use of a G-adherent transformation matrix W(k) which is written in the form (4.9). Suppose that for some index j, 1 <=j
(5.9)
Since the characteristic polynomial is supposed to be G-coherent and the transformation matrix is supposed to be G-adherent, we obtain by means of analytic continuation of (5.9) that X(k)w,(k)= 0 for every i, 1-< i<_m. This implies however that X(k)--O because the set {wj(k)} establishes a base for IKm(H;/~). Thus we have proved that either e ( j ) = 0 or ,(j)_~ 1 for every j, 1 <-j<=m. The possibility , ( j ) > 1 is ruled out because the eigenvalues of B(k) are distinct.
Holomorphic Matrices
353
For many physical applications it is of interest to know whether (5.1) has a solution which is holomorphic on G and positive-definite for almost every k e G c~ R,. Before giving our result concerning this problem we prove the following auxiliary theorem. Theorem 12. Let D ~ G \ X be a polydisc centered at ~ G n R n. Suppose that on D n Rn the eigenvalues of B(k) are real. Further, let W(k) be a G-adherent transformation matrix which transforms B(k) in the canonical Jordan form. For W(k) we take the construction of Theorem 7; moreover, the matrix elements of W(k) are assumed to be integers of IK(D;/~). I.
The matrix W(k) W t(k) defined on D can be continued analytically into the whole domain G. This analytic continuation, which we denote by ( w w t ) ( k ) , is single-valued and holomorphic at every point of G. In other words, the matrix elements of ( W W t ) ( k ) belong to IK(G).
II. Let ~ 2 ~ ( G \ , ? ) n R , denote the set of points where all the eigenvalues of B(k) are real. Then on f2 the matrix ( W W t ) ( k ) is non-negative definite. III. Let f2t ~ G c~R~ denote the set of points where at least one of the eigenvalues of (WW*)(k) is equal to zero. The R,-Lebesgue measure of f21, is zero. IV. l f B(k) is a matrix polynomial, while G=C~, the matrix ( w w t ) ( k ) is a matrix polynomial too. Proof. Because of the properties we imposed on the chosen transformation matrix W(k), the a th component of its i th column has the form
= Y
q,
(5A0)
q=l
where the t/qi,(k ) are integers of the field IK(D; B), 2~(k) is the eigenvalue of B(k) which corresponds to the column w~(k). Since 2,(k) is real on D c~ Rn, the analytic conjugate of (5.10) on D is w*,(k)= X t/*,,(k) (2i(k)) q.
(5.11)
q=l
Since the r/q ,,(k) and t/*i,(k) are restrictions of integers in IK(G; B, B t) and 2,(k) is an eigenvalue of B(k), the functions (5.10) and (5.11) can be continued analytically into the whole domain G. Of course, in general this analytic continuation will not be single-valued on G. I.
On D the (~, fl)th matrix element of the product
can be written
( w w t ) . a ( k ) = ~ w,~(k)w.*,p(k). i=l
(5.12)
354
J. DE GRAAF:
We choose the points PsD and Q~G\Z. Let/'1, F2cG\Z be paths which connect P and Q. Suppose that analytic continuation of (WW*),p(k) along /'1 and F2 yields two distinct holomorphic function elements at Q. This would
E Q$$
ifold Fig. 2
imply that analytic continuation of (wwt)~(k) from P to Q along F1 and then back from Q to P along/'2 yields a holomorphic function element at P which is different from (WWt)~a(k). However, since W(k) is G-adherent, analytic continuation of (WWt),p(k) along some closed path may change only the order of summation in (5.12) and therefore it leaves (wwt),p(k) unaltered. From this we conclude that the analytic continuation of (wwt)~ ~(k) is single valued and holomorphic on G. (The possible singularities at ~ can be removed (ef. BOCI-INER[11], p. 173, Th. 5).) II. On R, c~D the matrix Wt(k) equals the ordinary Hermitean transpose of W(k); therefore the matrix (wwt)(k) is non-negative definite on R, c~D. Let D' be a polydisc centered at some point x~f2, such that D'cG\r, and D' c~R, c t2. Let F be a path connecting D and D'. Analytic continuation of (5.10) and (5.11) along F shows that the continuation of w*,(k) into D' equals 2"
i
i
Fig. 3
the analytic continuation in D' of the continuation of w~(k) into D'. It is important to remark that this would not be the case if the analytic continuation of ;t~(k) were not real on D'c~R,. Obviously the continuation along F into D' of W* (k) is equal to the analytic transpose (in D') of the continuation of W(k) into D'. Therefore we conclude that also on R, r3D' W*(k) equals the ordinary Hermitean transpose of W(k) which implies that WW* is nonnegative on R, c3D'. III. The determinant of W(k) does not vanish identically on G. We conclude that also det[WW*] does not vanish identically on G. Det [WW*]=0 only at the points k where one of the eigenvalues of WW t is zero. Since det [WWt] is holomorphic on G, the set of its zeros is a thin subset of G. From the Appendix it follows that the intersection of this thin subset and the real manifold R, has R,-Lebesgue measure zero.
Holomorphic Matrices
355
IV. In this case ( w w t ) ( k ) is holomorphic everywhere in C., and at infinity its matrix elements do not grow faster than some finite power of [kl. Then by invoking the argument of the proof of part II of Theorem 4 it is easily seen that the matrix elements of ( W W t) (k) must be polynomials.
Example" In Cl we c~
the matrix ( -k2i ~i). Its eigenvalues are 21 2=
+_V k 2 - 1. We cut the k-plane and choose an interval I as indicated in the figure.
I
Re k
Fig. 4
Let f+ (k) and f _ (k) be the eigenvalues which are equal to + 1/k-~-----1-1and - 1/k-2-'Z-1-1 respectively, on L Then for W(k) we can choose
W
1 =(-2i(k+f+)
SO
(11 2i(k+f+)l wt=
and
2i(k+f_)/'
-2i(k+f_)) 1 W W ' = 2 \{2ik
2ik I 4,J"
8 k 2 --
The eigenvalues of this matrix are
~1, 2 = (8 k 2 - 3) + 1/(8 k 2 - 3) 2 - 1 6 ( k 2 - 1 ) . These are both positive only for k < - I 22 (k) are both real.
and k > 1, thus only when 21(k) and
Theorem 13.
I.
The necessary and sufficient conditions that (5.1) has a solution X(k) with the properties (a) X(k) is holomorphic everywhere on G; (b) X(k)> 0 on G c~R., while the equality sign may hoM true only on a subset of G n R. which has R.-Lebesgue measure zero are (a) there exists a polydisc D centered at some point ~ ( G \ Z ) n R . such that on D the matrix B(k) is similar to a diagonal matrix; (fl) on GeaR, the eigenvalues 2j(k) of B(k) are all reaL II. lf B(k) is a matrix polynomial on C,, property (a) of I can be replaced by (a') X(k) is a matrix polynomial. III. If, in addition, the eigenvalues of B(k) are non-negative on G n R . , and if X(k) is a solution of (5.1) having the properties (a)-(b), then X(k)B(k)>O for every k~G c~R..
356
J. DE GRAAF:
Proof. I. II. Sufficiency. We make use of the matrix (wwt)(k) as constructed in Theorem 12. Since in the present case the eigenvalues of B(k) are real everywhere on G c~R, (see Theorem 12 II-IV), we have (i) The eigenvalues of (wwt)(k) are strictly positive on the set E = ((G\Z) c~R,)\f21, i.e. almost everywhere on G c~R,. (ii) d e t [ W W t ] > 0 on E. Now we define X (k) = det [(W W t ) (k)]. ( W W t )- 1.
(5.13)
Property (a), or (a'), is easily verified by observing that, possibly up to a factor - 1 , the matrix elements of X(k) are sub-determinants of the matrix WW t. Concerning property (b), combination of (i) and (ii) yields the result that X(k) must be strictly positive definite on E. Since E is dense everywhere in G n R, (cfi Appendix), we conclude that X(k)>O everywhere on G c~R,. Further, on the polydisc D we have B(k)= W ( k ) A ( k ) w - l ( k ) where A(k) is a diagonal matrix whose elements 2g(k) are real on Dc~R,. Therefore A (k)= A t (k) on D. Finally, because on D,
( w w t ) - I B=(WWt) -1 WAW -1 = wt-IAW-I_-_ w t - ~ A t W t W t-1 W-1 = w t - I A t w t ( w w t ) -1 = B t ( w w t ) -1, the matrix X(k) defined by (5.13) is a solution of (5.1).
Necessity. Let there be given a solution X(k) of (5.1) which has the properties (a)-(b). Again let f21cGc~R . denote the set of points where det[X(k)]=0. At any ~E(G~Rn)\~21 we denote by X~(~) the Hermitean non-negative definite square root of X(~). On (Gc~R,)\f21 the inverse X-~(~) exists; it is also Hermitean and non-negative definite. Now at any ~e(Gc~R,)\t21 (5.1) can be written x ~ (~) B (~) x - * (~) = x - * (~) B t (~) x ~ (~).
Obviously X~(~)B(~)X-i(~) is Hermitean, so it is similar to a real diagonal matrix. But then, since X~BX -89 is similar to B, B(~) must also be similar to a real diagonal matrix at every ~e(Gc~R,)\f21. Now let D ~ G \ Z be a polydisc centered at xe((G\Z)nR,)\g21. On D we may write by use of Theorem 6 B(k) = W(k)J(k) W - 1(k) where J(k) is a Jordan matrix. Suppose that J(k) is not a diagonal matrix; then there exists no point keD where B(k) is similar to a diagonal matrix. This assumption leads to a contradiction because at the points of D n R, B(k) actually is similar to a diagonal matrix. III. Let ~ be a point of E. At ~ we can write B(~)=W(~)A(~)W-I(~) where A(~) is a non-negative diagonal matrix. From (5.1) we obtain
WtX WW- ~BW= w t B t w - ~t w t x w , wt x w A =AWt XW"
(5.14)
Holomorphic Matrices
357
Since X ( 0 is non-negative W t X W is also non-negative. Further, (5.14) shows that A and w t x w c o m m u t e ; therefore we conclude that W * X W A > O (cf. PERLIS [5], p. 213, Th. 9-33). But then also W t-~ W t X W A W -1 = X B = B t X > O at ~. Finally, because E is dense everywhere in GeaR,, (see the Appendix), and the matrix elements of X ( k ) B ( k ) are continuous, we obtain X B > 0 everywhere on G c~ R , .
Appendix Theorem. Let f ( k ) be a holomorphic function o f n complex variables on a connected
open domain G c C , . Suppose that f ( k ) does not vanish identically. Further, let S denote the set of points k e G where f ( k ) = 0 . Thus S is a thin subset of G. Then (i) The set S n Rn is closed in Rn. (ii) The set S c~ R, has R,-Lebesgue measure zero. (iii) The set R , \ S is dense everywhere in R,.
Proof. (i) S and Rn are both closed sets in C~. (ii) We give the p r o o f by means of induction. In the case n = 1, f ( k ) can be zero only at a set S consisting of isolated points. Thus R 1 n S consists only of isolated points. It is well k n o w n that the R1-Lebesgue measure of this set is zero. N o w assuming that the theorem is true if n = N, we shall prove it for n= N + 1. Since f does not vanish identically, there exist fixed numbers k~ . . . . . k~v+ 1 such thatf(k~ . . . . . k~v+x) =~0. F r o m the fact thatf(k~ . . . . , k~, k s + x) is a h o l o m o r p h i c function of the variable kN + a we conclude thatf(k~, ..., k~v, ks+ a) can be zero only at a set of isolated points of the real ks+ 1-axis. Therefore f ( k l , ..., ks, ks+l) considered as a h o l o m o r p h i c function of kl, ..., k s can be identically zero only for a set of isolated real numbers ks+ a. Thus almost every ks+a-section of S n R N + a has RN-Lebesgue measure zero. The conclusion is that the R s + i - L e b e s g u e measure of S n R N + a must be zero. See HALMOS [13], p. 137, Th. A.
References 1. BROER,L. J. F., & L. A. PELETIER, Some comments on linear wave equations. Appl. Sci. Res. 17, 65 (1967). 2. BROER,L. J. F., & L. A. PELETIER,On a class of conservative waves. Appl. Sci. Res. 17, 133 (1967). 3. BROER,L. J. F., & L. A. PELETIER, Some properties of the solutions of wave equations. Appl. Sei. Res. 21, 138 (1969). 4. DE GRAAF,J., Linear Dynamical Systems in Hilbert Space (Part I). Dissertation Technische Hogeschool Eindhoven (The Netherlands) 1970. 5. PERLIS,S., Theory of Matrices. Addison-Wesley Publ. Comp. Inc. 1958. 6. VANDER WAERDEN, B. L., Algebra I. Berlin-G6ttingen-Heidelberg: Springer 1955. 7. GUNNING,R. C., & H. Rosst, Analytic Functions of Several Complex Variables. Englewood Cliffs, N.J.: Prentice-Hall, Inc. 1965. 8. ~mov, G. E., Konecnomernye linejnye prostranstva. Isd. Nauka. Moskva 1969. 9. P.~IF~N,H. J., et al., Algebra, Hochschultaschenbficher 110/110a. Mannheim: Bibliographisches Institut.
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J. DE GRAAF: Holomorphic Matrices
10. FUKS, B. A., Introduction to the Theory of Analytic functions of Several Complex Variables. Am. Math. Soc. Providence, Rhode Island 1963. 11. BOCHNER,S., & W. T. MARTIN, Several Complex Variables. Princeton: University Press 1948. 12. GANTMACHER,F. R., Matrizenrechnung I. Berlin: Deutscher Veflag der Wissenschaften 1965. 13. HALMOS,P. R., Measure Theory. New York: D. Van Nostrand 1950. 14. DE GRAAF,J., & L. J. F. BROER, Plane waves in linear homogeneous media. Three papers, Reports on Mathematical Physics, 1972 3 (1), 3 (2), 3 (2). 15. HOCrdNC, J. G., & G. S. YOUNG, Topology. Addison-Wesley Publ. Comp., Inc. The Technological University Eindhoven
(Received October 25, 1971)
