Informally a n×n matrix A=(a _i,j ) is called a band matrix if all matrix elements vanish outside a diagonally bordered "band" of some range and size: a_i,j0 −k₁ < i − j < k₂ for some k₁, k₂ > 0. The width of the band is k₁ + k₂ − 1. A band matrix with k₁ = k₂ = 1 is a diagonal matrix; a band matrix with k₁ = k₂ = 2 is a tridiagonal matrix. If one puts k₁ = 1, k₂ = n, one obtains the definition of a lower triangular matrix, for k₁ = n, k₂ = 1 an upper triangular matrix. In numerical analysis and computing the notion band matrix also denotes a special type of matrix representation that uses two dimensional matrices as building blocks. It is used to store the result of a LU factorization. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In linear algebra, a LU decomposition, or LUP decomposition is a matrix decomposition of a matrix into a lower triangular matrix L, an upper-triangular matrix U, and a permutation matrix P. This decomposition is used in numerical analysis to solve systems of linear equations. ...

Specific examples for band matrices are:

• Diagonal matrix
• Tridiagonal matrix
• Upper and lower triangular matrix
• Upper and lower Hessenberg matrix

Some forms of band matrices are known as block matrix. In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ... In linear algebra, a tridiagonal matrix is one that is almost diagonal. ... In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ... In linear algebra, a Hessenberg matrix is one that is almost triangular. ... In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...

Note: The representation of the LAPACK Fortran package is different from that of EISPACK. Linear Algebra PACKage (LAPACK). ... Fortran (also FORTRAN) is a statically typed, compiled programming language originally developed in the 1950s and still heavily used for scientific computing and numerical computation half a century later. ... EISPACK has been superceded mostly by LAPACK. Categories: Software stubs ...

External links

• http://www.netlib.org/lapack/lug/node124.html Source of this information
• http://www.intel.com/software/products/mkl/docs/mklqref/matrixst.htm Overview of matrix representation
• http://www.cs.ut.ee/~toomas_l/linalg/lin1/node13.html An overview of band representations