Editing Toeplitz matrix (section)

{{Short description|Matrix with shifting rows}}
In [[linear algebra]], a '''Toeplitz matrix''' or '''diagonal-constant matrix''', named after [[Otto Toeplitz]], is a [[matrix (mathematics)|matrix]] in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix:

:<math>\qquad\begin{bmatrix}
a & b & c & d & e \\
f & a & b & c & d \\
g & f & a & b & c \\
h & g & f & a & b \\
i & h & g & f & a 
\end{bmatrix}.</math>

Any <math>n \times n</math> matrix <math>A</math> of the form

:<math>A = \begin{bmatrix}
  a_0 & a_{-1}   & a_{-2} & \cdots & \cdots & 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} & \cdots & \cdots & a_2    & a_1    & a_0
\end{bmatrix}</math>

is a '''Toeplitz matrix'''. If the <math>i,j</math> element of <math>A</math> is denoted <math>A_{i,j}</math> then we have
:<math>A_{i,j} = A_{i+1,j+1} = a_{i-j}.</math>

A Toeplitz matrix is not necessarily [[Square matrix|square]].