Zobrazit v EDS

Combinatorial identities deriving from the \(n\)th power of a \(2\times 2\) matrix

Uloženo v:
Podrobná bibliografie
Název: Combinatorial identities deriving from the \(n\)th power of a \(2\times 2\) matrix
Autoři: McLaughlin, James
Informace o vydavateli: University of West Georgia, Carrollton, GA; Charles University, Prague; DIMATIA
Témata: Matrix exponential and similar functions of matrices, Combinatorial identities, bijective combinatorics
Popis: Summary: We give a new formula for the \(n\)th power of a \(2\times 2\) matrix. More precisely, we prove the following: Let \(A=\left(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix}\right)\) be an arbitrary \(2\times 2\) matrix, \(T=a+d\) its trace, \(D=ad-bc\) its determinant and define \[ y_n:= \sum^{\lfloor n/2\rfloor}_{i=0}{n-i \choose i}T^{n-2i}(-D)^i. \] Then, for \(n\geq 1\), \[ A^n=\left(\begin{matrix} y_n-dy_{n-1} & by_{n-1}\\ cy_{n-1} & y_n-ay_{n-1}\end{matrix}\right). \] We use this formula together with an existing formula for the \(n\)th power of a matrix, various matrix identities, formulae for the \(n\)th power of particular matrices, etc. to derive various combinatorial identities.
Druh dokumentu: Article
Popis souboru: application/xml
Přístupová URL adresa: https://zbmath.org/2124091
Přístupové číslo: edsair.c2b0b933574d..0bb7fe8feb1367b4355776f78c3800bf
Databáze: OpenAIRE
Popis
Abstrakt:Summary: We give a new formula for the \(n\)th power of a \(2\times 2\) matrix. More precisely, we prove the following: Let \(A=\left(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix}\right)\) be an arbitrary \(2\times 2\) matrix, \(T=a+d\) its trace, \(D=ad-bc\) its determinant and define \[ y_n:= \sum^{\lfloor n/2\rfloor}_{i=0}{n-i \choose i}T^{n-2i}(-D)^i. \] Then, for \(n\geq 1\), \[ A^n=\left(\begin{matrix} y_n-dy_{n-1} & by_{n-1}\\ cy_{n-1} & y_n-ay_{n-1}\end{matrix}\right). \] We use this formula together with an existing formula for the \(n\)th power of a matrix, various matrix identities, formulae for the \(n\)th power of particular matrices, etc. to derive various combinatorial identities.