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# pentadiagonal matrix

An $n\times n$ pentadiagonal matrix (with $n\geq 3$) is a matrix of the form

$\begin{pmatrix}c_{1}&d_{1}&e_{1}&0&\cdots&\cdots&0\\ b_{1}&c_{2}&d_{2}&e_{2}&\ddots&&\vdots\\ a_{1}&b_{2}&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&a_{2}&\ddots&\ddots&\ddots&e_{{n-3}}&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&d_{{n-2}}&e_{{n-2}}\\ \vdots&&\ddots&a_{{n-3}}&b_{{n-2}}&c_{{n-1}}&d_{{n-1}}\\ 0&\cdots&\cdots&0&a_{{n-2}}&b_{{n-1}}&c_{n}\end{pmatrix}.$ |

It follows that a pentadiagonal matrix is determined by five vectors: one $n$-vector $c=(c_{1},\ldots,c_{n})$, two $(n-1)$-vectors $b=(b_{1},\ldots,b_{{n-1}})$ and $d=(d_{1},\ldots,d_{{n-1}})$, and two $(n-2)$-vectors $a=(a_{1},\ldots,a_{{n-2}})$ and $e=(e_{1},\ldots,e_{{n-2}})$. It follows that a pentadiagonal matrix is completely determined by $n+2(n-1)+2(n-2)=5n-6$ scalars.

Keywords:

pentadiagonal penta-diagonal

Related:

TridiagonalMatrix

Synonym:

penta-diagonal matrix

Type of Math Object:

Definition

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## Mathematics Subject Classification

15-00*no label found*65-00

*no label found*

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new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

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new question: A trascendental number. by Ron Castillo

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