Tridiagonal matrix

In linear algebra, a tridiagonal matrix is a matrix that is "almost" a diagonal matrix. To be exact: a tridiagonal matrix has nonzero elements only in the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.

For example, the following matrix is tridiagonal::

A determinant formed from a tridiagonal matrix is known as a continuant. [cite book | author=Thomas Muir | authorlink=Thomas Muir (mathematician) | title=A treatise on the theory of determinants | date=1960 | publisher=Dover Publications | pages=516-525 ]

Properties

A tridiagonal matrix is of Hessenberg type. Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have one of these properties. Furthermore, if a real tridiagonal matrix "A" satisfies "a"_"k","k"+1 "a"_"k"+1,"k" > 0, so that the signs of its entries are symmetric, then it is similar to a Hermitian matrix, and hence, its eigenvalues are real. The latter conclusion continues to hold if we replace the condition "a"_"k","k"+1 "a"_"k"+1,"k" > 0 by "a"_"k","k"+1 "a"_"k"+1,"k" ≥ 0.

The set of all "n × n" tridiagonal matrices form a "3n-2"
dimensional vector space.

Many linear algebra algorithms require significantly less computational effort when applied to diagonal matrices, and this improvement often carries over to tridiagonal matrices as well. For instance, the determinant of a tridiagonal matrix "A" of order "n" can be computed by the recursive formula for a continuant:$det\; A\; =\; a\_\{n,n\}\; det\; ,\; [A]\; \_\{\{1,ldots,n-1\; -\; a\_\{n,n-1\}\; a\_\{n-1,n\}\; det\; ,\; [A]\; \_\{\{1,ldots,n-2\; ,\; ,,$where det $[A]\; \_\{\{1,ldots,k$ denotes the "k"th principal minor, that is, $[A]\; \_\{\{1,ldots,k$ is the submatrix formed by the first "k" rows and columns of "A". The cost of computing the determinant of a tridiagonal matrix using this formula is linear in "n", while the cost is cubic for a general matrix.

Computer programming

A transformation that reduces a general matrix to Hessenberg form will reduce a Hermitian matrix to "tridiagonal" form. Thus, many eigenvalue algorithms, when applied to a Hermitian matrix, reduce the input Hermitian matrix to tridiagonal form as a first step.

A "tridiagonal matrix" can also be stored more efficiently than a general matrix by using a special 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" − 1 containing the subdiagonal and superdiagonal elements.

A system of tridiagonal matrix $A\; x\; =\; b$, for $bin\; eals^n$ can be solved by a specific algorithm requiring "O(n)" operations (Golub and Van Loan).

References

* Roger A. Horn and Charles R. Johnson, "Matrix Analysis," Cambridge University Press, 1985. ISBN 0-521-38632-2.
* Gene H. Golub and Charles F. Van Loan, "Matrix Computations (3rd Edt.)," Johns Hopkins Univ Pr., 1996. ISBN 0-8018-5414-8.
* Bianca Beatriz Banagudos, Katherine Guerrero, and Donna Fe Gagaracruz, "Mathematics for the New Millennium," Regional Science High School for R-IX, 2008-2009, IV-Quisumbing. ISBN 0-1234-5678-9.

External links

* [http://www.netlib.org/lapack/lug/node125.html Tridiagonal and Bidiagonal Matrices] in the LAPACK manual.
* [http://math.fullerton.edu/mathews/n2003/Tri-DiagonalMod.html Module for Tri-Diagonal Linear Systems]