Toeplitz matrices

E451536

matrix class structured matrix

Toeplitz matrices are structured matrices whose entries are constant along each diagonal, playing a central role in operator theory, numerical analysis, and signal processing.

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Observed surface forms (2)

Surface form	Occurrences
Toeplitz matrix	0
Slepian sequences	1

Statements (50)

Predicate	Object
instanceOf	matrix class ⓘ structured matrix ⓘ
appearsIn	Toeplitz systems of linear equations ⓘ autoregressive (AR) model covariance matrices ⓘ discretization of integral equations ⓘ
hasAlgorithm	Bareiss algorithm for Toeplitz systems NERFINISHED ⓘ Durbin algorithm NERFINISHED ⓘ Levinson recursion NERFINISHED ⓘ superfast Toeplitz solver ⓘ
hasApplication	deconvolution in signal processing ⓘ design of FIR filters ⓘ fast solution of Yule–Walker equations ⓘ fast solution of linear prediction problems ⓘ
hasComplexity	O(n log^2 n) superfast methods (typical) ⓘ O(n^2) direct solution methods ⓘ
hasProperty	O(n) parameters for an n×n matrix ⓘ constant entries along each diagonal ⓘ determined by first row and first column ⓘ each descending diagonal from left to right is constant ⓘ entry a_{i,j} depends only on i-j ⓘ generally not diagonalizable by Fourier transform ⓘ non-generic eigenvalue distribution ⓘ often non-normal ⓘ
hasSpecialCase	Hermitian Toeplitz matrix ⓘ banded Toeplitz matrix ⓘ block Toeplitz matrix ⓘ block Toeplitz matrix with Toeplitz blocks ⓘ circulant matrix ⓘ symmetric Toeplitz matrix ⓘ tri-diagonal Toeplitz matrix ⓘ
namedAfter	Otto Toeplitz NERFINISHED ⓘ
relatedTo	Hankel matrix ⓘ Laurent operator ⓘ Szegő limit theorem NERFINISHED ⓘ Toeplitz operator NERFINISHED ⓘ Wiener–Hopf factorization ⓘ convolution ⓘ discrete-time linear time-invariant system ⓘ
subclassOf	constant-diagonal matrix ⓘ discrete convolution operator matrix ⓘ
usedIn	control theory ⓘ filter design ⓘ image processing ⓘ numerical analysis ⓘ numerical linear algebra ⓘ operator theory ⓘ signal processing ⓘ spectral estimation ⓘ system identification ⓘ time series analysis ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Gábor Szegő → fieldOfWork → Toeplitz matrices ⓘ

Carathéodory–Fejér interpolation → uses → Toeplitz matrices ⓘ

David Slepian → notableWork → Toeplitz matrices ⓘ

this entity surface form: Slepian sequences