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,…}:

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:

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:

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

Visualize the entries of the Toeplitz matrix: