Compact Abelian Semigroups of Matrices

Abstract:

      The eigenvalues of the matrices in a compact abelian [Hermite] matrix semigroup are determined by means of the simultaneous diagonalization or trianglization of matrices. It is shown that the eigenvalues of each matrix of a compact semigroup consisting of invertible Hermite matrices are $\pm 1$, that a simple semigroup of Hermite matrices is similar to a semigroup of diagonal matrices, and that the spectrum radium of a compact commutative semigroup of matrices is less than or equal to $1$, etc.