Pairs of Matrices with Property L(0

1. It was proved by Frobenius [l] that any function f(A, B) of two commutative «X« square matrices A, B has as characteristic roots the numbers /(X¿, Hi) where X¿ are the characteristic roots of A and pi the characteristic roots of B, both taken in a special ordering independent of the function /. However, commutativity of A and B is known not to be a necessary condition [2]. Matrices which have the above property are said to have property P. Recently M. Kac suggested a study of matrices A, B of a less restricted nature, namely those for which any linear combination aA +ßB has as characteristic roots the numbers a\i+ßpi. In general such matrices do not have property P. This paper is concerned with the study of such matrices which will be called matrices with property L (for linear). It will be shown, confirming a conjecture of M. Kac, that hermitian matrices with property L are commutative. If n = 2, property L implies property P. The paper, being concerned with pairs of matrices, concludes with an enumeration of seven properties of pairs of matrices. Each of these properties is implied by its successor. It is shown that already for matrices with w = 3 these properties are in general nonequivalent. Wherever there is no mention of hermitian matrices the results hold for matrices whose elements instead of being complex numbers belong to an algebraically closed field with arbitrary characteristic.