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:

Define a function for constructing a circulant matrix from a vector:

Multiplying a circulant matrix with a vector can be expressed as a cyclic convolution:

Multiplying a vector with a circulant matrix can be expressed as a cyclic correlation:

Circulant matrices can be diagonalized by the Fourier matrix:

The diagonal elements of the resulting diagonal matrix are the same as the product of the Fourier matrix and the starting vector, up to a constant scaling factor:

A Toeplitz matrix with symbolic entries:

The inverse of a Toeplitz matrix can be determined from just the first and last column of the inverse. Use LinearSolve to compute the first column of the inverse:

Compute the last column of the inverse:

The Gohberg–Semencul formula for the inverse of a Toeplitz matrix uses the first and last columns to assemble the full inverse matrix:

Verify the inverse:

Define a tridiagonal Toeplitz matrix:

Show the 4×4 case:

Verify an expression for the characteristic polynomial in terms of the Chebyshev polynomial of the second kind for the first few cases:

A tridiagonal Toeplitz matrix can be diagonalized by a diagonal rescaling of the type 1 discrete sine transform matrix:

Define the Kac–Murdock–Szegő (KMS) matrix as a ToeplitzMatrix:

The KMS matrix is a symmetric Toeplitz matrix:

The KMS matrix is the correlation matrix of an autoregressive process of order one (i.e. an AR(1) process):

The inverse of the KMS matrix is a tridiagonal matrix:

The characteristic polynomial of the KMS matrix can be expressed in terms of the Chebyshev polynomial of the second kind:

Define the Parter matrix as a ToeplitzMatrix:

The Parter matrix is both a Toeplitz matrix and a Cauchy matrix:

The Parter matrix's dominant singular values are clustered around :

The prolate matrix is a symmetric positive-definite Toeplitz matrix that appears in multitaper power spectral density estimation:

Visualize the entries of the prolate matrix:

The condition number increases exponentially with n:

The 2-norm condition number is the ratio of largest to smallest eigenvalue due to symmetry:

The daily exchange rates of the euro to the dollar from May 2012 through September 2012:

Visualize the data:

Fit an order-3 autoregressive process (ARProcess) to the exchange rates by solving the Yule-Walker equations:

The same result can be obtained from EstimatedProcess:

The order Padé approximant of a function about the point x=x₀ can be obtained from the power series coefficients. Define the function, the numerator and denominator degrees, and the expansion point:

Generate the coefficients of the power series up to order :

The coefficients of the denominator can be obtained by solving a Toeplitz system constructed from the power series coefficients:

The coefficients of the numerator can be obtained by multiplying the vector of denominator coefficients with a Toeplitz matrix constructed from the power series coefficients:

Assemble the Padé approximant:

Compare with the result of PadeApproximant:

Generate the first few signed elementary symmetric polynomials in variables:

Generate the first few power sums in variables:

Verify the Newton–Girard formulas [MathWorld] relating power sums and elementary symmetric polynomials:

Define the Ching matrix as a ToeplitzMatrix:

The Ching matrix is a lower Hessenberg 0-1 matrix (a matrix with entries that are either zero or one, and is zero above the first superdiagonal):

The determinant of the order n Ching matrix is equal to the n Fibonacci number, and gives an upper bound for the determinant of all lower Hessenberg 0-1 matrices:

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: