Tridiagonal matrix

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In linear algebra, a tridiagonal matrix is one that is "almost" diagonal. 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:

$\begin{pmatrix} 1 & 4 & 0 & 0 \\ 3 & 4 & 1 & 0 \\ 0 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 \\ \end{pmatrix}$

is tridiagonal.

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

$\det A = a_{n,n} \det \, [A]_{\{1,\ldots,n-1\}} - a_{n,n-1} a_{n-1,n} \det \, [A]_{\{1,\ldots,n-2\}} \, ,\,$

where det $[A]_{\{1,\ldots,k\}}$ denotes the kth 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.

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, involve reduction 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 $A x = b\in \reals^n$ can be solved by a specific algorithm requiring O(n) operations (Golub and Van Loan).

[edit] See also

Triangular matrix

[edit] 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.