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:

[\begin1 & 4 & 0 & 0 \\3 & 4 & 1 & 0 \\0 & 2 & 3 & 4 \\0 & 0 & 1 & 3 \\\end]

is tridiagonal.

A tridiagonal matrix is Hessenberg. 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.

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_ \det \, [A]_} - a_ a_ \det \, [A]_} \, ,\, ]

where det [[A]_}] denotes the kth principal minor, that is, [[A]_}] 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 algorithm requiring 8n floating point 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 0801854148.

External links

[Tridiagonal and Bidiagonal Matrices] in the LAPACK manual.

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