Manuals/calci/TOEPLITZ

MATRIX("TOEPLITZ",order)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle order} is the size of the Toeplitz matrix.

Description

This function gives the matrix of order 3 with the property of toeplitz matrix.
A Toeplitz matrix is a matrix with the constant values along negative sloping diagonals(descending diagonal from left to right).
If the i,j element of A is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{i,j}} , then we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{i,j} = A_{i+1,j+1} = a_{i-j}} .

Any nxn matrix A of the form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} a_0 & a_{-1} & a_{-2} & \cdots & a_{-n+1} \\ a_1 & a_0 & a_{-1} & \cdots \\ a_2 & a_1 & \cdots \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} \\ a_{1} & a_0 & a_{-1} & \cdots & \ddots & & \vdots \\ a_{2} & a_{1} & \cdots& \ddots & \ddots & \ddots& \vdots \\ \vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2}\\ \vdots & & \ddots & a_{1} & a_{0}& a_{-1} \\ a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \end{bmatrix} }

The property of Toeplitz matrix is :Toeplitz matrices are persymmetric.
Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric.
Toeplitz matrices commute asymptotically.

Examples

MATRIX("toeplitz",15,1..10)

A = \begin{bmatrix}

 a_{0} & a_{-1} & a_{-2} & \ldots & \ldots  &a_{-n+1}  \\
 a_{1} & a_0  & a_{-1} &  \ddots   &  &  \vdots \\
 a_{2}    & a_{1} & \ddots  & \ddots & \ddots& \vdots \\ 
\vdots &  \ddots & \ddots &   \ddots  & a_{-1} & a_{-2}\\
\vdots &         & \ddots & a_{1} & a_{0}&  a_{-1} \\

a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \end{bmatrix}