Math. Zeitschr. 71, 463--465 (1959)

A w e a k p r o p e r t y L for pairs o f matr ices By

OLGA TAUSSKY

A pair of n • n matrices A, B with complex numbers as elements is said to have the L-property if 2 A + # B has as characteristic roots 2a i+#f l i , where a~i are the characteristic roots of A and fib the characteristic roots of B - - t a k e n in a special o rder ing- -and when 2,/~ run through all complex numbers. Let one of the matrices, say A, be diagonable (i.e. similar to a diagonal matrix). The following condition on B is then necessary for A and B to have property L (see T. S. MOTZXI~ and O. TAUSSKY [11 p. 108--t09).

Let t be the number o[ different characteristic roots o / A and assume that all the characteristic roots o~ i o] A are arranged in sets of equal ones. Let m i be the multiplicity o[ o~ i and fli the corresponding characteristic, root o/B. Let B* = P-1B P when A * = P-1A P is in diagonal/orm. Then

B * = B~I 22 " . B2t

B t l Bt~ ... Btt

where Bii is :an mi • mi matrix, i = t . . . . . t, and

(1) ix1 - B.I = n ( x - s

where r , are the m i characteristic roots corresponding to ~i and where I is the unit matrix.

Let us denote this condi t ion 'as "Proper ty l", since it is weaker than property L in general.

THEOREVt t. Property l implies property L for n = 2.

PROOF. For n = 2 we have two cases here: firstly, the case of A a scalar matrix, which certainly implies property L; secondly the case where A has two different roots, In this case the two diagonal elements of B* are its characteristic roots. This implies that B* is triangular, hence A* and B*, and then also A and B have property L .

THEORE~ 2. Property l implies commutativity /or normal matrices when the trans]orming matrix P is taken as unitary.

PROOF. For normal matrices it is known (see N. WIEGMANN [21, PAR- KER [3~) that under the condition (1) all Bil,=O in B* for i ~ k . Since all t3~i are normal too, each of them can be transformed into diagonal form by a similarity which leaves the corresponding scalar matrix in A* unchanged.

464 OLC.A TAUSSKY :

Hence A and B can be transformed to diagonal form simultaneously, hence not only have t~roperty L, but even commute.

That proper ty l plays some role is shown by Theorem 3 which is the main result of this note.

THEOREM 3' Let A, B be a pair o / n • matrices With complex numbers as elements. Let A be diagonable. Let o~,: be the characteristic roots o/ A and ~i(z) be the characteristic roots o/ A + zB where z is a complex variable. I]

_.dz4~!_

(the characteristic roots o / B ) /or some ordering, then A and B have property l.

PROOF. Assume A already in diagonal form. This is no restriction. Let ~i have multiplici ty m i. Among the n (not necessarily different) functions ~i(z) exactly m i will have the proper ty tha t ?i(0) : ~ i . All ~i(z) satisfy the determinantal equation

(2) ]A + B z - - I~(z)[ = 0,

To obtain ~(z) for those i for which ~,i(0)=x~ we differentiate equation (2) m i times. When differentiating the left hand side we have to form the sum of all determinants in which m i rows of (2)--not necessarily , different ones-- have been differentiated. Any row which contains our 0q is of the form

zb, x, zb,., . . . . . ~ i+ z b , , - - ? i ( z ) . . . . . zb,,,.

This whole row vanishes for z = O. Hence only the one determinant in which all these mi rows have been differentiated comes in question for z = O. ~All the other rows in this determinant have then not been differentiated. For simplicity let at i correspond to the first m i rows. T h e n for z = 0

O =

b ~ - - ~,~(0) b~ b21 b~ ~ -- ~,~ (0)

b m~ I b ,ni 2

0 0

0 0

b l m ~ . . . . . . . b l n

�9 ' �9 ~ ~ * ~ ~ b2~r

�9 . . . . . . .

b ~ , , . , - r ~ ( o ) . . . . . . . b . , .

0 ~ m t + l - - ~ X i 0

0 . . . . . . . . 0

0 0 . . . . . . *on ~ ~

This shows tha t m i of the y~(O) are charac te r i s t i c roo ts of the first m~xm~ pr inc ipa l minor of B and similar facts hold for the other ~i. On the other hand, each y~(0) is equal to some of the fli by assumption. This proves Theorem 3. The converse is true too provided the derivatives exist 1).

THEOREM 4. For n = 2 any two matrices A, B /or which y~ (0) (as defined in Theorem 3) is equal to fli /or some ordering have property L.

l] This shows that property l is an infinitesimal property L at z = O.

A weak property L for pairs of matrices 465

PROOF. If A is d iagonable then this follows i m m e d i a t e l y from Theorems t

a n d 3 . There only remains the case where A is s imi lar t o a m a t r i x ( ; 1).

In this case we m a y res t r i c t ourselves to the case where A is a l r e a d y in th is form. The m a t r i x A § Bz is then

( ~ + b l l z t +b:2z~ b21z ~+b22z]

I t s e igenvalues are

~i (Z) = (~ -31- g bll -[- b22 -~- V~-(b l l - - 52 2) 3 -t- 4 521 2: (t + 512 Z)/2. 2

F o r b21=0 the mat r ices A and B have a l r ea dy p r o p e r t y L. F o r b2 :4=0 we h a v e y ; ( 0 ) = e~, hence this case does not come unde r our a s sumpt ion .

THEOREM 5- Two normal matrices A and B / o r which ~ (0) (as de/ined in Theorem 3) is equal to fli in some ordering commute.

PROOF. This follows i m m e d i a t e l y from Theorems 2 a n d 3 since eve ry normal m a t r i x is un i t a r i l y s imi lar to a d iagona l ma t r ix .

REMARK. A special case of no rma l �9 ma t r i ces a r e he rmi t i an matr ices . F o r he rmi t i an mat r ices i t is known (RELLICH [4]) t h a t the e igenvalues of A + Bz can be expanded into a power series for a ne ighborhood of the origin. Hence

7~(z) = ~ i + q : z + q 2 z 2 + ' " .

Theorem 5 shows t h a t ci:=fli impl ies t h a t ci2=qa . . . . . 0.

R e f e r e n c e s

[1] MOTZKIN, T. S., and O. TAussKY: Pairs of matrices with property L. Trans. Amer. Math. Soc. 73, 108-- 1 t4 (1952). -- [2] WIEGMANN, N.: Normal matrices with property L. Proc. Amer. Math. Soc. 4 , 35--36 (t953). -- [3] PARKER, W. V.: A note on normal matrices. Amer. Math. Monthly 61, 330--33t (1954). -- [4] RELLICH, F.: St6rungstheorie der Spektralzerlegung. I. Math. Annalen 113, 600--619 (t936). -- [5]'~VIELANDT, ~I,: Pairs of normal matrices with property L. J. Res. Nat. Bur. Standards 51, 89--90 (1953).

Dept. o/Math., Cali/ornia Institute o/ Technology, Pasadena, Cal. (U.S.A.)

(Eingegangen am 18. Miirz 1959)