A weak property L for pairs of matrices

Abstract

A pair of n x n matrices A, B with complex numbers as elements is said to have the L-property if λA + µB has as characteristic roots λα_i + µβ_i, where α_i are the characteristic roots of A and β_i the characteristic roots of B--taken in a special ordering--and when λ,µ 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. MOTZKIN and O. TAUSSKY [1] p. 108--109).

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