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

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Titel: Combinatorial identities deriving from the \(n\)th power of a \(2\times 2\) matrix
Autoren: McLaughlin, James
Verlagsinformationen: University of West Georgia, Carrollton, GA; Charles University, Prague; DIMATIA
Schlagwörter: Matrix exponential and similar functions of matrices, Combinatorial identities, bijective combinatorics
Beschreibung: 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.
Publikationsart: Article
Dateibeschreibung: application/xml
Zugangs-URL: https://zbmath.org/2124091
Dokumentencode: edsair.c2b0b933574d..0bb7fe8feb1367b4355776f78c3800bf
Datenbank: OpenAIRE
Beschreibung
Abstract: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.