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Toeplitz matrix

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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:

[ 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}}.}

Any n × n {\displaystyle n\times n} matrix A {\displaystyle A} of the form

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}}}

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

A i , j = A i + 1 , j + 1 = a i − j . {\displaystyle A_{i,j}=A_{i+1,j+1}=a_{i-j}.}

A Toeplitz matrix is not necessarily square.

Solving a Toeplitz system

A matrix equation of the form

A x = b {\displaystyle Ax=b}

is called a Toeplitz system if A {\displaystyle A} is a Toeplitz matrix. If A {\displaystyle A} is an n × n {\displaystyle n\times n} Toeplitz matrix, then the system has at most only 2 n − 1 {\displaystyle 2n-1} unique values, rather than n 2 {\displaystyle n^{2}} . 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 algorithms such as the Schur algorithm or the Levinson algorithm in O ( n 2 ) {\displaystyle O(n^{2})} time. Variants of the latter have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). The algorithms can also be used to find the determinant of a Toeplitz matrix in O ( n 2 ) {\displaystyle O(n^{2})} time.

A Toeplitz matrix can also be decomposed (i.e. factored) in O ( n 2 ) {\displaystyle O(n^{2})} 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.

Properties

  • An n × n {\displaystyle n\times n} Toeplitz matrix may be defined as a matrix A {\displaystyle A} where A i , j = c i − j {\displaystyle A_{i,j}=c_{i-j}} , for constants c 1 − n , … , c n − 1 {\displaystyle c_{1-n},\ldots ,c_{n-1}} . The set of n × n {\displaystyle n\times n} Toeplitz matrices is a subspace of the vector space of n × n {\displaystyle n\times n} matrices (under matrix addition and scalar multiplication).
  • Two Toeplitz matrices may be added in O ( n ) {\displaystyle O(n)} time (by storing only one value of each diagonal) and multiplied in O ( n 2 ) {\displaystyle O(n^{2})} 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.
  • For symmetric Toeplitz matrices, there is the decomposition
1 a 0 A = G G T − ( G − I ) ( G − I ) T {\displaystyle {\frac {1}{a_{0}}}A=GG^{\operatorname {T} }-(G-I)(G-I)^{\operatorname {T} }}
where G {\displaystyle G} is the lower triangular part of 1 a 0 A {\displaystyle {\frac {1}{a_{0}}}A} .
A − 1 = 1 α 0 ( B B T − C C T ) {\displaystyle A^{-1}={\frac {1}{\alpha _{0}}}(BB^{\operatorname {T} }-CC^{\operatorname {T} })}
where B {\displaystyle B} and C {\displaystyle C} are lower triangular Toeplitz matrices and C {\displaystyle C} is a strictly lower triangular matrix.

Discrete convolution

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 {\displaystyle h} and x {\displaystyle x} can be formulated as:

y = h ∗ x = [ h 1 0 ⋯ 0 0 h 2 h 1 ⋮ ⋮ h 3 h 2 ⋯ 0 0 ⋮ h 3 ⋯ h 1 0 h m − 1 ⋮ ⋱ h 2 h 1 h m h m − 1 ⋮ h 2 0 h m ⋱ h m − 2 ⋮ 0 0 ⋯ h m − 1 h m − 2 ⋮ ⋮ h m h m − 1 0 0 0 ⋯ h m ] [ x 1 x 2 x 3 ⋮ x n ] {\displaystyle y=h\ast x={\begin{bmatrix}h_{1}&0&\cdots &0&0\\h_{2}&h_{1}&&\vdots &\vdots \\h_{3}&h_{2}&\cdots &0&0\\\vdots &h_{3}&\cdots &h_{1}&0\\h_{m-1}&\vdots &\ddots &h_{2}&h_{1}\\h_{m}&h_{m-1}&&\vdots &h_{2}\\0&h_{m}&\ddots &h_{m-2}&\vdots \\0&0&\cdots &h_{m-1}&h_{m-2}\\\vdots &\vdots &&h_{m}&h_{m-1}\\0&0&0&\cdots &h_{m}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\end{bmatrix}}}
y T = [ h 1 h 2 h 3 ⋯ h m − 1 h m ] [ x 1 x 2 x 3 ⋯ x n 0 0 0 ⋯ 0 0 x 1 x 2 x 3 ⋯ x n 0 0 ⋯ 0 0 0 x 1 x 2 x 3 … x n 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 ⋯ 0 0 x 1 ⋯ x n − 2 x n − 1 x n 0 0 ⋯ 0 0 0 x 1 ⋯ x n − 2 x n − 1 x n ] . {\displaystyle y^{T}={\begin{bmatrix}h_{1}&h_{2}&h_{3}&\cdots &h_{m-1}&h_{m}\end{bmatrix}}{\begin{bmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}&0&0&0&\cdots &0\\0&x_{1}&x_{2}&x_{3}&\cdots &x_{n}&0&0&\cdots &0\\0&0&x_{1}&x_{2}&x_{3}&\ldots &x_{n}&0&\cdots &0\\\vdots &&\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\cdots &0&0&x_{1}&\cdots &x_{n-2}&x_{n-1}&x_{n}&0\\0&\cdots &0&0&0&x_{1}&\cdots &x_{n-2}&x_{n-1}&x_{n}\end{bmatrix}}.}

This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.

Infinite Toeplitz matrix

Main article: Toeplitz operator

A bi-infinite Toeplitz matrix (i.e. entries indexed by Z × Z {\displaystyle \mathbb {Z} \times \mathbb {Z} } ) A {\displaystyle A} induces a linear operator on ℓ 2 {\displaystyle \ell ^{2}} .

A = [ ⋮ ⋮ ⋮ ⋮ ⋯ a 0 a − 1 a − 2 a − 3 ⋯ ⋯ a 1 a 0 a − 1 a − 2 ⋯ ⋯ a 2 a 1 a 0 a − 1 ⋯ ⋯ a 3 a 2 a 1 a 0 ⋯ ⋮ ⋮ ⋮ ⋮ ] . {\displaystyle A={\begin{bmatrix}&\vdots &\vdots &\vdots &\vdots \\\cdots &a_{0}&a_{-1}&a_{-2}&a_{-3}&\cdots \\\cdots &a_{1}&a_{0}&a_{-1}&a_{-2}&\cdots \\\cdots &a_{2}&a_{1}&a_{0}&a_{-1}&\cdots \\\cdots &a_{3}&a_{2}&a_{1}&a_{0}&\cdots \\&\vdots &\vdots &\vdots &\vdots \end{bmatrix}}.}

The induced operator is bounded if and only if the coefficients of the Toeplitz matrix A {\displaystyle A} are the Fourier coefficients of some essentially bounded function f {\displaystyle f} .

In such cases, f {\displaystyle f} is called the symbol of the Toeplitz matrix A {\displaystyle A} , and the spectral norm of the Toeplitz matrix A {\displaystyle A} coincides with the L ∞ {\displaystyle L^{\infty }} norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 of.

See also

  • Circulant matrix, a square Toeplitz matrix with the additional property that a i = a i + n {\displaystyle a_{i}=a_{i+n}}
  • Hankel matrix, an "upside down" (i.e., row-reversed) Toeplitz matrix
  • Szegő limit theorems – Determinant of large Toeplitz matrices
  • Toeplitz operator – compression of a multiplication operator on the circle to the Hardy spacePages displaying wikidata descriptions as a fallback

Notes

  1. Press et al. 2007, §2.8.2—Toeplitz matrices
  2. Hayes 1996, Chapter 5.2.6
  3. Krishna & Wang 1993
  4. Monahan 2011, §4.5—Toeplitz systems
  5. Brent 1999
  6. Bojanczyk et al. 1995
  7. Mukherjee & Maiti 1988
  8. Böttcher & Grudsky 2012

References

Further reading

Matrix classes
Explicitly constrained entries
Constant
Conditions on eigenvalues or eigenvectors
Satisfying conditions on products or inverses
With specific applications
Used in statistics
Used in graph theory
Used in science and engineering
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