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Every unit matrix is a LULU

โœ Scribed by Gilbert Strang

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
288 KB
Volume
265
Category
Article
ISSN
0024-3795

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โœฆ Synopsis

The four matrices LoUoL1U1 at the end of the title are triangular with ones on their main diagonals. Their product has determinant one. Following a question and theorem of Toffoli, we show that any matrix with determinant one can be factored in this way. A transformation of the plane becomes a sequence of one-dimensional shears, with n 2 -1 free parameters.

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We show that every n ร‚ n complex nonderogatory matrix is similar to a unique unit upper Hessenberg Toeplitz matrix. The proof is constructive, and can be adapted to nonderogatory matrices with entries in any ยฎeld of characteristic zero or characteristic greater than n. We also prove that every n ร‚ n

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