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

  • J. K. Goldhaber (a1)

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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 f1j, 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.

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References

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1. Albert, A. A., Structure of Algebras (New York, 1946).
2. Fettis, H. E.,A method for obtaining the characteristic equation of a matrix and computing the associated modal columns, Quarterly J. Appl. Math., vol 8 (1950), 206212.
3. Frame, J. S., A simple recursion formula for inverting a matrix, Abstract 471, Bull. Amer. Math. Soc, vol. 55 (1949), 1045.
4. Frobenius, G., Über vertauschbare Matrizen, Sitz. preuss. Akad. Wiss. (1896), 601614.
5. MacDuffee, C. C., The theory of matrices, Ergeb. der Math., vol. 2 (1933).
6. MacDuffee, C. C., Vectors and matrices, Carus Mathematical Monograph no. 7 (1943).
7. McCoy, N. H., On the characteristic roots of matric polynomials, Bull. Amer. Math. Soc, vol. 42 (1936), 592600.
8. Rademacher, H., On a theorem of Frobenius, Studies and Essays presented to R. Courant (New York, 1948), 301305.
9. Williamson, J., The simultaneous reduction of two matrices to triangular form, Amer. J. Math., vol. 57 (1935), 281293.
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The Homomorphic Mapping of Certain Matric Algebras onto Rings of Diagonal Matrices

  • J. K. Goldhaber (a1)

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