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

Source abstract

The problem of determining the conditions under which a finite set of matrices A 1 A 2 , … , 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 ( A 1 A 2 … , A k ) in these matrices has characteristic roots f (λ 1j , λ 2 j …,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.