Матрица Тёплица

Матрица Тёплица (диагонально-постоянная матрица) — матрица, в которой на всех диагоналях, параллельных главной, стоят равные элементы:

A={\begin{bmatrix}a_{0}&a_{-1}&a_{-2}&\ldots &\ldots &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}&\ldots &\ldots &a_{2}&a_{1}&a_{0}\end{bmatrix}}

то есть выполняется соотношение:

a_{i,\,j}=a_{i-1,\,j-1}

Названы в честь немецкого математика Отто Тёплица.

Пример

Матрица 4×5:

M={\begin{pmatrix}4&5&6&7&8\\3&4&5&6&7\\2&3&4&5&6\\1&2&3&4&5\\\end{pmatrix}}

Свойства

Две матрицы Тёплица можно сложить за $O(n)$ операций. Матрицу Тёплица можно умножить на вектор за $O(n\log n)$ операций, а умножение матриц Тёплица можно провести за $O(n^{2})$ операций.

Тёплицева система линейных уравнений, то есть система вида $Ax=b$ , где $A$ — тёплицева матрица, может быть решена методом Левинсона за время $O(n^{2})$ ^[1]^[2].

Матрицы Тёплица также связаны с рядами Фурье: оператор умножения на многочлен из синусов или косинусов, спроецированный на конечномерное пространство, можно представить такой матрицей.

См. также

Примечания

↑ Krishna H., Wang Y.. The Split Levinson Algorithm is Weakly Stable (англ.) // SIAM Journal on Numerical Analysis^[англ.]. — 1993. — Vol. 30, iss. 5. — P. 1498—1508. — doi:10.1137/0730078.
↑ Блейхут Р. Э.. Быстрые алгоритмы цифровой обработки сигналов / Пер. с англ. И. И. Грушко. — М.: Мир, 1989. — 448 с. — ISBN 5-09-001009-2.

Литература

Тыртышников Е. Е. Теплицевы матрицы, некоторые их аналоги и приложения / отв. ред. чл.-корр. АН СССР В. В. Воеводин. — М.: ВИНИТИ, 1989. — 184 с. Архивировано 26 августа 2017 года.
Gray R. M.. Toeplitz and Circulant Matrices: A Review // Foundations and Trends in Communications and Information Theory. — 2006. — Vol. 2, № 3. — P. 155—239. — doi:10.1561/0100000006.
Бабенко К. И. О тёплицевых и ганкелевых матрицах // Успехи математических наук. — 1986. — Т. 41, вып. 1(247). — С. 171—178.
Иохвидов И. С. Ганкелевы и тёплицевы матрицы и формы. Алгебраическая теория. — М.: Наука, 1974.
Пустыльников Л. Д.. Тёплицевы и ганкелевы матрицы и их применения // Успехи математических наук. — 1984. — Т. 39, вып. 4(238). — С. 53—84.

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