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

Toeplitz matrices arise in various problems, including analysis of time series, discretization of constant coefficient differential equations, and discretization of convolution equations .

Since a Toeplitz matrix depends on just parameters it is reasonable to expect that a linear system can be solved in less than the flops that would be required by LU factorization. Indeed methods are available that require only flops; see Golub and Van Loan (2013) for details.

Upper triangular Toeplitz matrices can be written in the form

where is upper bidiagonal with a superdiagonal of ones and . It follows that the product of two upper triangular Toeplitz matrices is again upper triangular Toeplitz, upper triangular Toeplitz matrices commute, and is also an upper triangular Toeplitz matrix (assuming is nonzero, so that is nonsingular).

Tridiagonal Toeplitz matrices arise frequently:

The eigenvalues of are

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

It has a number of interesting properties.

In MATLAB, a Toeplitz matrix can be constructed using `toeplitz(c,r)`

, which produces the matrix with first column `c`

and first row `r`

. Example:

>> 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

## References

- Gene Golub and Charles F. Van Loan, Matrix Computations, fourth edition, Johns Hopkins University Press, Baltimore, MD, USA, 2013. Section 4.7.

## Related Blog Posts

- What Is a Circulant Matrix? (2022)
- What Is a Tridiagonal Matrix? (2022)
- What Is the Kac–Murdock–Szegö Matrix? (2020)

This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.