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
2 … , A
k) in these matrices has characteristic roots f(λ1j, λ2j
…,λki) (j = 1, 2, … , n) was first considered by Frobenius . 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  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.