# set4 - \$ Set 4 Factoring Toeplitz Matrices Kyle A Gallivan...

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a39 a38 a36 a37 Set 4: Factoring Toeplitz Matrices Kyle A. Gallivan Department of Mathematics Florida State University Numerical Linear Algebra 1 Fall 2010 1

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a39 a38 a36 a37 Toeplitz and Schur Complement Let n = 4 and consider the symmetric Toeplitz matrix T = τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 4 τ 1 τ 2 τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 3 τ 1 τ 2 τ 2 1 τ 1 τ 2 τ 1 τ 4 τ 1 τ 3 τ 1 τ 2 τ 2 1 . 2
a39 a38 a36 a37 One Step of Cholesky Let n = 4 . If the Cholesky factor was Toeplitz then the first step would define a Toeplitz L . L = τ 1 0 0 0 τ 2 τ 1 0 0 τ 3 τ 2 τ 1 0 τ 4 τ 3 τ 2 τ 1 . This would be an O ( n ) algorithm!! 3

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a39 a38 a36 a37 Check the Factorization LL T = τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 4 −− τ 2 1 + τ 2 2 τ 1 τ 2 + τ 2 τ 3 τ 1 τ 3 + τ 2 τ 4 −− −− τ 2 1 + τ 2 2 + τ 2 3 τ 1 τ 2 + τ 2 τ 3 + τ 3 τ 4 −− −− −− τ 2 1 + τ 2 2 + τ 2 3 + τ 2 4 . and we have LL T = T + 0 0 0 0 −− τ 2 2 τ 2 τ 3 τ 2 τ 4 −− −− τ 2 2 + τ 2 3 τ 2 τ 3 + τ 3 τ 4 −− −− −− τ 2 2 + τ 2 3 + τ 2 4 4
a39 a38 a36 a37 Displacement Definition 4.1. Let Z = parenleftBig e 2 e 3 e 4 . . . e n 0 parenrightBig = 0 T e T 1 e T 2 . . . e T n - 1 and M R n × n . The displacement of M with respect to Z is M = M ZMZ T Note that Z is nilpotent since Z n = 0 . 5

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a39 a38 a36 a37 Toeplitz and Displacement T = T ZTZ T = τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 4 τ 1 τ 2 τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 3 τ 1 τ 2 τ 2 1 τ 1 τ 2 τ 1 τ 4 τ 1 τ 3 τ 1 τ 2 τ 2 1 0 0 0 0 0 τ 2 1 τ 1 τ 2 τ 1 τ 3 0 τ 1 τ 2 τ 2 1 τ 1 τ 2 0 τ 1 τ 3 τ 1 τ 2 τ 2 1 = τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 4 τ 1 τ 2 0 0 0 τ 1 τ 3 0 0 0 τ 1 τ 4 0 0 0 6
a39 a38 a36 a37 Toeplitz and Displacement T = τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 4 τ 1 τ 2 0 0 0 τ 1 τ 3 0 0 0 τ 1 τ 4 0 0 0 = τ 1 0 τ 2 τ 2 τ 3 τ 3 τ 4 τ 4 1 0 0 1 τ 1 τ 2 τ 3 τ 4 0 τ 2 τ 3 τ 4 Rank 2 matrix, trivially computable for symmetric Toeplitz. This is not true for all low-rank displacement M . 7

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a39 a38 a36 a37 Toeplitz and Schur Complement The Schur complement with respect to the 1 , 1 element is T sc = τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 2 τ 2 1 τ 1 τ 2 τ 1 τ 3 τ 1 τ 2 τ 2 1 τ 2 τ 3 τ 4 parenleftBig τ 2 τ 3 τ 4 parenrightBig = τ 2 1 τ 2 2 τ 1 τ 2 τ 2 τ 3 τ 1 τ 3 τ 2 τ 4 τ 1 τ 2 τ 3 τ 2 τ 2 1 τ 2 3 τ 1 τ 2 τ 3 τ 4 τ 1 τ 3 τ 4 τ 2 τ 1 τ 2 τ 4
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