The problem of determining the conditions under which a finite set of matrices A
1A2, … , A
k has the property that their characteristic roots λ1j, λ2j, … , λki (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(λ1j, λ2j
…,λki) (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.