ToeplitzMatrix

ToeplitzMatrix[n]

gives the n×n Toeplitz matrix with first row and first column being successive integers.

ToeplitzMatrix[{c1,c2,…,cn}]

gives the Toeplitz matrix whose first column consists of elements c1, c2, ….

ToeplitzMatrix[{c1,c2,…,cm},{r1,r2,…, rn}]

gives the Toeplitz matrix with elements ci down the first column, and ri across the first row.

Details

  • The element r1 must be the same as c1. »

Examples

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Basic Examples  (3)

4×4 Toeplitz matrix:

Toeplitz matrix with first column {c1,…} and first row {c1,r2,…}:

Scope  (5)

Machine-number Toeplitz matrix:

20-digit-precision Toeplitz matrix:

Toeplitz matrices with complex entries:

Nonsquare Toeplitz matrices:

A common symbolic notation for Toeplitz matrices:

Properties & Relations  (4)

Size-20 Toeplitz matrix:

ToeplitzMatrix[{c1,c2,…}] is Hermitian if c1 is real:

has all real eigenvalues:

is diagonalizable by a unitary matrix:

ToeplitzMatrix[c,r] is a circulant matrix when r=RotateRight[Reverse[c]]:

The eigenvalues can be found from:

where is the Fourier matrix and is DiagonalMatrix[v]:

HankelMatrix and ToeplitzMatrix are related by reversed identity matrix:

Possible Issues  (1)

When r1 is not the same as c1, the value of c1 is used and r1 ignored:

Neat Examples  (1)

Wolfram Research (2007), ToeplitzMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/ToeplitzMatrix.html.

Text

Wolfram Research (2007), ToeplitzMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/ToeplitzMatrix.html.

CMS

Wolfram Language. 2007. "ToeplitzMatrix." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ToeplitzMatrix.html.

APA

Wolfram Language. (2007). ToeplitzMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ToeplitzMatrix.html

BibTeX

@misc{reference.wolfram_2022_toeplitzmatrix, author="Wolfram Research", title="{ToeplitzMatrix}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ToeplitzMatrix.html}", note=[Accessed: 03-June-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_toeplitzmatrix, organization={Wolfram Research}, title={ToeplitzMatrix}, year={2007}, url={https://reference.wolfram.com/language/ref/ToeplitzMatrix.html}, note=[Accessed: 03-June-2023 ]}