What Is a Toeplitz Matrix? – Nick Higham

is a Toeplitz matrix if for parameters . A Toeplitz matrix has fixed diagonals. For :

Toeplitz matrices come up in numerous issues, together with evaluation of time sequence, discretization of fixed coefficient differential equations, and discretization of convolution equations .

Since a Toeplitz matrix relies on simply parameters it’s affordable to anticipate {that a} linear system might be solved in lower than the flops that will be required by LU factorization. Certainly strategies can be found that require solely flops; see Golub and Van Mortgage (2013) for particulars.

Higher triangular Toeplitz matrices might be written within the type

the place is higher bidiagonal with a superdiagonal of ones and . It follows that the product of two higher triangular Toeplitz matrices is once more higher triangular Toeplitz, higher triangular Toeplitz matrices commute, and can also be an higher triangular Toeplitz matrix (assuming is nonzero, in order that is nonsingular).

Tridiagonal Toeplitz matrices come up regularly:

The eigenvalues of are

The Kac–Murdock–Szegö matrix is the symmetric Toeplitz matrix

It has a variety of fascinating properties.

In MATLAB, a Toeplitz matrix might be constructed utilizing toeplitz(c,r), which produces the matrix with first column c and first row r. Instance:

>> n = 5; A = toeplitz(1:n,[1 -2:-1:-n])
A =
     1    -2    -3    -4    -5
     2     1    -2    -3    -4
     3     2     1    -2    -3
     4     3     2     1    -2
     5     4     3     2     1