In the mathematical discipline of linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix:
Any n×n matrix A of the form
is a Toeplitz matrix. If the i,j element of A is denoted Ai,j, then we have
Generally, a matrix equation
is the general problem of n linear simultaneous equations to solve. If A is an Toeplitz matrix, then the system is rather special (has only m+n-1 degrees of freedom, rather than m n). One could therefore expect that solution of a Toeplitz system would be easier.
This can be investigated by the transformation
which has rank 2, where Uk is the down-shift operator. Specifically, one can by simple calculation show that
where empty places in the matrix are zeros.
The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of h and x can be formulated as:
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