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

Published online by Cambridge University Press:  20 November 2018

J. K. Goldhaber
J. K. Goldhaber*
Affiliation:
University of Wisconsin
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The problem of determining the conditions under which a finite set of matrices A1A2, … , Ak has the property that their characteristic roots λ1j, λ2j, … , λki (j = 1, 2, …, n) may be so ordered that every polynomial f(A1A2 … , Ak) 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 (Ai〉 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 31 - 42
Copyright
Copyright © Canadian Mathematical Society 1952

References

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