In 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
A Toeplitz matrix is not necessarily square.
Solving a Toeplitz systemEdit
A matrix equation of the form
is called a Toeplitz system if A is a Toeplitz matrix. If A is an Toeplitz matrix, then the system has only 2n−1 degrees of freedom, rather than n2. We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.
Toeplitz systems can be solved by the Levinson algorithm in Θ(n2) time. Variants of this algorithm have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). The algorithm can also be used to find the determinant of a Toeplitz matrix in O(n2) time.
A Toeplitz matrix can also be decomposed (i.e. factored) in O(n2) time. The Bareiss algorithm for an LU decomposition is stable. An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.
- An n×n Toeplitz matrix may be defined as a matrix A where Ai,j = ci−j, for constants c1−n … cn−1. The set of n×n Toeplitz matrices is a subspace of the vector space of n×n matrices under matrix addition and scalar multiplication.
- Two Toeplitz matrices may be added in O(n) time (by storing only one value of each diagonal) and multiplied in O(n2) time.
- Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric.
- Toeplitz matrices are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix. Similarly, one can represent linear convolution as multiplication by a Toeplitz matrix.
- Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity.
- A positive semi-definite n×n Toeplitz matrix of rank r < n can be uniquely factored as
- where is an r×r positive definite diagonal matrix, is an n×r Vandermonde matrix such that the columns are . Here and is normalized frequency, and is the Hermitian transpose of . If the rank r = n, then the Vandermonde decomposition is not unique.
- For symmetric Toeplitz matrices, there is the decomposition
- where is the lower triangular part of .
- The inverse of a nonsingular symmetric Toeplitz matrix has the representation
- where and are lower triangular Toeplitz matrices and is a strictly lower triangular matrix.
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 and can be formulated as:
Infinite Toeplitz matrixEdit
A bi-infinite Toeplitz matrix (i.e. entries indexed by ) induces a linear operator on .
The induced operator is bounded if and only if the coefficients of the Toeplitz matrix are the Fourier coefficients of some essentially bounded function .
In such cases, is called the symbol of the Toeplitz matrix , and the spectral norm of the Toeplitz matrix coincides with the norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 in the google book link: 
- Press et al. 2007, §2.8.2—Toeplitz matrices
- Krishna & Wang 1993
- Monahan 2011, §4.5—Toeplitz systems
- Brent 1999
- Bojanczyk et al. 1995 harvnb error: no target: CITEREFBojanczykBrentHoogSweet1995 (help)
- Stewart 2003
- Chen, Hurvich & Lu 2006
- Chan & Jin 2007
- Chandrasekeran et al. 2007
- Yang, Xie & Stoica 2016
- Mukherjee & Maiti 1988
- Böttcher & Grudsky 2012
- Bojanczyk, A. W.; Brent, R. P.; de Hoog, F. R.; Sweet, D. R. (1995), "On the stability of the Bareiss and related Toeplitz factorization algorithms", SIAM Journal on Matrix Analysis and Applications, 16: 40–57, arXiv:1004.5510, doi:10.1137/S0895479891221563
- Böttcher, Albrecht; Grudsky, Sergei M. (2012), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, ISBN 978-3-0348-8395-5
- Brent, R. P. (1999), "Stability of fast algorithms for structured linear systems", in Kailath, T.; Sayed, A. H. (eds.), Fast Reliable Algorithms for Matrices with Structure, SIAM, pp. 103–116
- Chan, R. H.-F.; Jin, X.-Q. (2007), An Introduction to Iterative Toeplitz Solvers, SIAM
- Chandrasekeran, S.; Gu, M.; Sun, X.; Xia, J.; Zhu, J. (2007), "A superfast algorithm for Toeplitz systems of linear equations", SIAM Journal on Matrix Analysis and Applications, 29 (4): 1247–1266, CiteSeerX 10.1.1.116.3297, doi:10.1137/040617200
- Chen, W. W.; Hurvich, C. M.; Lu, Y. (2006), "On the correlation matrix of the discrete Fourier transform and the fast solution of large Toeplitz systems for long-memory time series", Journal of the American Statistical Association, 101 (474): 812–822, CiteSeerX 10.1.1.574.4394, doi:10.1198/016214505000001069
- Krishna, H.; Wang, Y. (1993), "The Split Levinson Algorithm is weakly stable", SIAM Journal on Numerical Analysis, 30 (5): 1498–1508, doi:10.1137/0730078
- Monahan, J. F. (2011), Numerical Methods of Statistics, Cambridge University Press
- Mukherjee, Bishwa Nath; Maiti, Sadhan Samar (1988), "On some properties of positive definite Toeplitz matrices and their possible applications" (PDF), Linear Algebra and Its Applications, 102: 211–240, doi:10.1016/0024-3795(88)90326-6
- Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), Numerical Recipes: The Art of Scientific Computing (Third ed.), Cambridge University Press, ISBN 978-0-521-88068-8
- Stewart, M. (2003), "A superfast Toeplitz solver with improved numerical stability", SIAM Journal on Matrix Analysis and Applications, 25 (3): 669–693, doi:10.1137/S089547980241791X
- Yang, Zai; Xie, Lihua; Stoica, Petre (2016), "Vandermonde decomposition of multilevel Toeplitz matrices with application to multidimensional super-resolution", IEEE Transactions on Information Theory, 62 (6): 3685–3701, arXiv:1505.02510, doi:10.1109/TIT.2016.2553041
- Bareiss, E. H. (1969), "Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices", Numerische Mathematik, 13 (5): 404–424, doi:10.1007/BF02163269
- Goldreich, O.; Tal, A. (2018), "Matrix rigidity of random Toeplitz matrices", Computational Complexity, 27 (2): 305–350, doi:10.1007/s00037-016-0144-9
- Golub G. H., van Loan C. F. (1996), Matrix Computations (Johns Hopkins University Press) §4.7—Toeplitz and Related Systems
- Gray R. M., Toeplitz and Circulant Matrices: A Review (Now Publishers)
- Noor, F.; Morgera, S. D. (1992), "Construction of a Hermitian Toeplitz matrix from an arbitrary set of eigenvalues", IEEE Transactions on Signal Processing, 40 (8): 2093–2094, Bibcode:1992ITSP...40.2093N, doi:10.1109/78.149978
- Pan, Victor Y. (2001), Structured Matrices and Polynomials: unified superfast algorithms, Birkhäuser, ISBN 978-0817642402
- Ye, Ke; Lim, Lek-Heng (2016), "Every matrix is a product of Toeplitz matrices", Foundations of Computational Mathematics, 16 (3): 577–598, arXiv:1307.5132, doi:10.1007/s10208-015-9254-z