Tridiagonal matrix

In linear algebra, a tridiagonal matrix is a matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.

For example, the following matrix is tridiagonal:

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

The determinant of a tridiagonal matrix is given by the continuant of its elements.

An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form can be done with the Lanczos algorithm.

Properties

A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. In particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q/2 = n -- the dimension of the tridiagonal. 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 for all k, so that the signs of its entries are symmetric, then it is similar to a Hermitian matrix, by a diagonal change of basis matrix. Hence, its eigenvalues are real. If we replace the strict inequality by a_k,k+1 a_k+1,k ≥ 0, then by continuity, the eigenvalues are still guaranteed to be real, but the matrix need no longer be similar to a Hermitian matrix.

The set of all n × n tridiagonal matrices forms 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.

Determinant

Main article: continuant (mathematics)

The determinant of a tridiagonal matrix A of order n can be computed from a three-term recurrence relation. Write f₁ = |a₁| = a₁ and

$f_n = \begin{vmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-1} \\ & & & c_{n-1} & a_n \end{vmatrix}.$

The sequence (f_i) is called the continuant and satisfies the recurrence relation

$f_n = a_n f_{n-1} - c_{n-1}b_{n-1}f_{n-2}$

with initial values f₀ = 1 and f_-1 = 0. 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.

Inversion

The inverse of a non-singular tridiagonal matrix T

$T = \begin{pmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-1} \\ & & & c_{n-1} & a_n \end{pmatrix}$

is given by

$(T^{-1})_{ij} = \begin{cases} (-1)^{i+j}b_i \cdots b_{j-1} \theta_{i-1} \phi_{j+1}/\theta_n & \text{ if } i \leq j\\ (-1)^{i+j}c_j \cdots c_{i-1} \theta_{j-1} \phi_{i+1}/\theta_n & \text{ if } i > j\\ \end{cases}$

where the θ_i satisfy the recurrence relation

$\theta_i = a_i \theta_{i-1} - b_{i-1}c_{i-1}\theta_{i-2} \quad \text{ for } i=2,3,\ldots,n$

with initial conditions θ₀ = 1, θ₁ = a₁ and the ϕ_i satisfy

$\phi_i = a_i \phi_{i+1} - b_i c_i \phi_{i+2} \quad \text{ for } i=n-1,\ldots,1$

with initial conditions ϕ_n+1 = 1 and ϕ_n = a_n.

Closed form solutions can be computed for special cases such as symmetric matrices with all off-diagonal elements equal or Toeplitz matrices and for the general case as well.

In general, the inverse of a tridiagonal matrix is a semiseparable matrix and vice versa.

Solution of linear system

Main article: tridiagonal matrix algorithm

A system of equations A x = b for $\scriptstyle b\in \reals^n$ can be solved by an efficient form of Gaussian elimination when A is tridiagonal called tridiagonal matrix algorithm, requiring O(n) operations.

Eigenvalues

When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, namely $a + 2 \sqrt{bc} \, \cos(k \pi / {(n+1)})$, for $k=1,...,n.$

A real symmetric tridiagonal matrix has real eigenvalues, and all the eigenvalues are distinct (simple) if all off-diagonal elements are nonzero. Numerous methods exist for the numerical computation of the eigenvalues of a real symmetric tridiagonal matrix to arbitrary finite precision, typically requiring $O(n^2)$ operations for a matrix of size $n\times n$, although fast algorithms exist which require only $O(n\ln n)$.

Computer programming

A transformation that reduces a general matrix to Hessenberg form will reduce a Hermitian matrix to tridiagonal form. So, 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.

See also

• Pentadiagonal matrix

External links

• Tridiagonal and Bidiagonal Matrices in the LAPACK manual.
• Module for Tri-Diagonal Linear Systems
• High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
• Tridiagonal linear system solver in C++