Manuals/calci/TOEPLITZ

MATRIX("TOEPLITZ",order)

• $order$ 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 $A_{i,j}$, then we have

$A_{i,j}=A_{i+1,j+1}=a_{i-j}$.

• Any nxn matrix A of the form:

${\begin{bmatrix}a_{0}&a_{-1}&a_{-2}&\cdots &a_{-n+1}\\a_{1}&a_{0}&a_{-1}&\cdots \\a_{2}&a_{1}&\cdots \\\vdots &\ddots &\vdots \\0&\cdots &0\end{bmatrix}}$ ${\begin{bmatrix}a_{0}&a_{-1}&a_{-2}&\ldots &\ldots &a_{-n+1}\\a_{1}&a_{0}&a_{-1}&\cdots &\ddots &\vdots \\a_{2}&a_{1}&\cdots &\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &\ddots &a_{-1}&a_{-2}\\\vdots &&\ddots &a_{1}&a_{0}&a_{-1}\\a_{n-1}&\ldots &\ldots &a_{2}&a_{1}&a_{0}\end{bmatrix}}$

• 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}