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Pairs of matrices with property L.

University of British Columbia

Let A and B be n-square complex matrices with eigenvalues λ₁, λ₂,… λn and μ₁, μ₂,…μn respectively. The matrices A and B are said to have property L if any linear combination aA + bB, with a, b complex, has as eigenvalues the numbers aλᵢ + bμᵢ, i = 1,2, …,n. A theorem of Dr. M. D. Marcus, which gives a necessary and sufficient condition such that two matrices A and B have property L in terms of the traces of various power-products of A and B, is proved. This theorem is used to investigate the conditions on B for the special cases n = 2, 3, and 4, when A is in Jordan canonical form. The final result is a theorem which gives a necessary condition on B for A and B to have property L when A is in Jordan canonical form.

Text

2012-01-19

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http://hdl.handle.net/2429/40177

Mathematics

University of British Columbia

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