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On the Jordan form of a family of linear mappings

✍ Scribed by C.W. Norman

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 530 KB
Volume: 257
Category: Article
ISSN: 0024-3795
DOI: 10.1016/s0024-3795(96)00155-3

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✦ Synopsis

Let A and B be square matrices over a field F having their eigenvalues A and /z in F, and let g(x, y) = E~,s ar, xry ~ be a polynomial over F. Assuming the Jordan forms of A and B to be known, the Jordan form of E~,~ a,.~A ~ ® B ~ is determined when the partial derivatives ag/dx and ag/c~y are nonzero at (A,/z), thus generalizing the classical cases g(x, y)= x + y and g(x, y)= xy. The particular cases g(x, y) = x k + yl and g(x, y) = xky l are also generalized by using properties of the partial Hasse derivatives of g at (A,/x). The case where A and B are 2 × 2 or 3 X 3 matrices, g being arbitrary, is discussed exhaustively.

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