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Toeplitz matrix
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<p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a Toeplitz matrix or diagonal-constant matrix, named after <a href="/facts/Otto_Toeplitz/rmsRb2uJ">Otto Toeplitz</a>, is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: </p> [ 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 ] . {\displaystyle \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}}.} <p>Any n × n {\displaystyle n\times n} matrix A {\displaystyle A} of the form </p> A = [ a 0 a − 1 a − 2 ⋯ ⋯ a − ( n − 1 ) a 1 a 0 a − 1 ⋱ ⋮ a 2 a 1 ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ a − 1 a − 2 ⋮ ⋱ a 1 a 0 a − 1 a n − 1 ⋯ ⋯ a 2 a 1 a 0 ] {\displaystyle 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}}} <p>is a Toeplitz matrix. If the i , j {\displaystyle i,j} element of A {\displaystyle A} is denoted A i , j {\displaystyle A_{i,j}} then we have </p> A i , j = A i + 1 , j + 1 = a i − j . {\displaystyle A_{i,j}=A_{i+1,j+1}=a_{i-j}.} <p>A Toeplitz matrix is not necessarily <a href="/facts/Square_matrix/5TP9mNNn">square</a>. </p>