Tridiagonal matrix: Difference between revisions

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{{Short description|Matrix with nonzero elements on the main diagonal and the diagonals above and below it}}

In [[linear algebra]], a '''tridiagonal matrix''' is a [[band matrix]] that has nonzero elements only on the [[main diagonal]], the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main diagonal).For example, the following [[matrix (mathematics)|matrix]] is [[Tridiagonal matrix algorithm|tridiagonal]]:

For example, the following [[matrix (mathematics)|matrix]] is [[Tridiagonal matrix algorithm|tridiagonal]]:

:<math>\begin{pmatrix}

1 & 4 & 0 & 0 \\

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==Computer programming==

A transformation that reduces a general matrix to Hessenberg form will reduce a Hermitian matrix to tridiagonal form. So, many [[eigenvalue algorithm]]s, when applied to a Hermitian matrix, reduce the input Hermitian matrix to (symmetric real) tridiagonal form as a first step.<ref>{{Cite journal |last=Eidelman |first=Yuli |last2=Gohberg |first2=Israel |last3=Gemignani |first3=Luca |date=2007-01-01 |title=On the fast reduction of a quasiseparable matrix to Hessenberg and tridiagonal forms |url=https://www.sciencedirect.com/science/article/pii/S0024379506003041 |journal=Linear Algebra and its Applications |language=en |volume=420 |issue=1 |pages=86–101 |doi=10.1016/j.laa.2006.06.028 |issn=0024-3795}}</ref>

A tridiagonal matrix can also be stored more efficiently than a general matrix by using a special [[matrix representation|storage scheme]]. For instance, the [[LAPACK]] [[Fortran]] package stores an unsymmetric tridiagonal matrix of order ''n'' in three one-dimensional arrays, one of length ''n'' containing the diagonal elements, and two of length ''n'' &minus; 1 containing the [[subdiagonal]] and [[superdiagonal]] elements.