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On the Latent Roots of Certain Matrices

Published online by Cambridge University Press:  20 January 2009

A. C. Aitken
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In the present note certain known theorems on the latent roots of matrices are deduced from the fundamental theorem that a matrix A can be expressed in the form PQP-1, where P is non-singular and Q has zero elements everywhere to the left of the principal diagonal, and the latent roots of A in the diagonal. [The presence or absence of non-zero elements to the right of the diagonal is known to depend on the nature of the “elementary divisors” of the “characteristic determinant” of A, but in what follows these will not concern us.]

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 1 , Issue 2 , March 1928 , pp. 135 - 138
Copyright
Copyright © Edinburgh Mathematical Society 1928

References

page 135 note 1 See Muir's, History of the Theory of Determinants, Vol. I, p. 121.Google Scholar

page 136 note 1 Zur Theorie der Adjungirten Substitutionen. Math. Ann. 48 (1897), 417–424.Google Scholar

page 137 note 1 On the characteristic equations of certain linear substitutions. Quart. Journ. Math. 33 (1902), 80–84.Google Scholar

page 138 note 1 See Muir's, History of the Theory of Determinants, Vol. II, pp. 52–53.Google Scholar

page 138 note 2 Note on Induced Linear Substitutions. Amer. Journ. Math. 16 (1894), pp. 205–206.CrossRefGoogle Scholar

page 138 note 3 A Determinantal Equation whose Roots are the Product of the Roots of Given Determinantal Equations. Proc. Roy. Soc. Edin. 38 (1917), 57–60.Google Scholar