# ToeplitzMatrix

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 :

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)

Introduced in 2007
(6.0)