Band matrix

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: ai,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.}

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.

Note: The representation of the LAPACK Fortran package is different from that of EISPACK.

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