The Homomorphic Mapping of Certain Matric Algebras onto Rings of Diagonal Matrices

Abstract

The problem of determining the conditions under which a finite set of matrices A₁A₂, … , A_k has the property that their characteristic roots λ_1j, λ_2j, … , λ_ki (j = 1, 2, …, n) may be so ordered that every polynomial f(A₁A₂ … , A_k) in these matrices has characteristic roots f(λ_1j, λ_2j …,λ_ki) (j = 1, 2, … , n) was first considered by Frobenius [4]. He showed that a sufficient condition for the (A_i〉 to have this property is that they be commutative. It may be shown by an example that this condition is not necessary.J. Williamson [9] considered this problem for two matrices under the restriction that one of them be non-derogatory. He then showed that a necessary and sufficient condition that these two matrices have the above property is that they satisfy a certain finite set of matric equations.