Built-in Mathematica Symbol

ToeplitzMatrix

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

ToeplitzMatrix[{c₁, c₂, ..., c_n}]
gives the Toeplitz matrix whose first column consists of elements c₁, c₂, ....

ToeplitzMatrix[{c₁, c₂, ..., c_m}, {r₁, r₂, ..., r_n}]
gives the Toeplitz matrix with elements c_i down the first column, and r_i across the first row.

MORE INFORMATION

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, ...}:

4×4 Toeplitz matrix:

In[1]:=

Out[1]//MatrixForm=

In[1]:=

Out[1]//MatrixForm=

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

In[1]:=

Out[1]//MatrixForm=

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:

t has all real eigenvalues:

t 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)