# 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 entries c1, c2, .

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

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

# Details and Options  • Toeplitz matrices typically occur in applications related to approximation theory, signal processing, statistics and time series.
• A Toeplitz matrix is a matrix that is constant along its diagonals. The entries of the Toeplitz matrix are given by if , and otherwise.
• • The entry r1 must be the same as c1. »
• ToeplitzMatrix[,TargetStructure->struct] returns the Toeplitz matrix in the format specified by struct. Possible settings include:
•  Automatic automatically choose the representation returned "Dense" represent the matrix as a dense matrix "Structured" represent the matrix as a structured array
• With , a dense matrix is returned if the number of matrix entries is less than a preset threshold, and a structured array is returned otherwise.
• For a structured ToeplitzMatrix sa, the following properties "prop" can be accessed as sa["prop"]:
•  "ColumnVector" vector of entries down the first column "RowVector" vector of entries across the first row "Properties" list of supported properties "Structure" type of structured array "StructuredData" internal data stored by the structured array "StructuredAlgorithms" list of functions with special methods for the structured array "Summary" summary information, represented as a Dataset
• Normal[ToeplitzMatrix[]] converts the structured Toeplitz matrix to an ordinary matrix.

# Examples

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## Basic Examples(3)

A 4×4 symmetric Toeplitz matrix:

A 4×4 symmetric Toeplitz matrix with symbolic entries:

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

## Scope(9)

Make a Toeplitz matrix of machine numbers:

Make a Toeplitz matrix with 20-digit precision:

Toeplitz matrices with complex entries:

Rectangular Toeplitz matrices:

A common symbolic notation for Toeplitz matrices:

Generate a structured Toeplitz matrix:

The structured representation typically uses much less memory:

ToeplitzMatrix objects include properties that give information about the array:

The "ColumnVector" property gives the first column of the Toeplitz matrix:

The "RowVector" property gives the first row of the Toeplitz matrix:

The "Summary" property gives a brief summary of information about the array:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

When appropriate, structured algorithms return another ToeplitzMatrix object:

The transpose is also a ToeplitzMatrix:

The product of a Toeplitz matrix and its transpose is no longer a Toeplitz matrix:

Convert a dense Toeplitz matrix to a structured Toeplitz matrix:

## Options(2)

### TargetStructure(2)

Return the Toeplitz matrix as a dense matrix:

Return the Toeplitz matrix as a structured array:

With the setting , a dense matrix is returned for small dimensions:

For large dimensions, a structured representation is returned:

## Properties & Relations(5)

The determinant of the Toeplitz matrix of size is :

Cyclic ListConvolve with zero-padding is equivalent to multiplication with a lower triangular Toeplitz matrix:

Cyclic ListCorrelate with zero-padding is equivalent to multiplication with an upper triangular Toeplitz matrix:

ToeplitzMatrix[{c1,c2,}] is Hermitian if c1 is real: has all real eigenvalues: is diagonalizable by a unitary matrix:

ToeplitzMatrix and HankelMatrix are related by multiplication with an exchange matrix (a reversed identity matrix):

Equivalently, reversing a Toeplitz matrix gives a Hankel matrix:

## Possible Issues(1)

When r1 is not the same as c1, the value of c1 is used and r1 ignored: ## Neat Examples(1)

Visualize the entries of the Toeplitz matrix: