Tridiagonal matrix

In linear algebra, a "tridiagonal matrix" is a band matrix that has nonzero elements on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal only.
For example, the following matrix is tridiagonal:
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 — 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.
The determinant of a tridiagonal matrix "A" of order "n" can be computed from a three-term recurrence relation. Write "f"1 = |"a"1| = "a"1 (i.e., "f"1 is the determinant of the 1 by 1 matrix consisting only of "a"1), and let
The sequence ("f""i") is called the continuant and satisfies the recurrence relation
with initial values "f"0 = 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"
is given by
where the "θi" satisfy the recurrence relation
with initial conditions "θ"0 = 1, "θ"1 = "a"1 and the "ϕ""i" satisfy
with initial conditions "ϕ""n"+1 = 1 and "ϕ""n" = "an".
Closed form solutions can be computed for special cases such as symmetric matrices with all diagonal and 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.
A system of equations "Ax" = "b" for formula_8 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 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 formula_10 operations for a matrix of size formula_11, although fast algorithms exist which (without parallel computation) require only formula_12.
As a side note, an unreduced symmetric tridiagonal matrix is a matrix containing non-zero off-diagonal elements of the tridiagonal, where the eigenvalues are distinct while the eigenvectors are unique up to a scale factor and are mutually orthogonal.
For "unsymmetric" tridiagonal matrices one can compute the eigendecomposition using a similarity transformation.
Similarity to symmetric tridiagonal matrix.
Given a real tridiagonal, "nonsymmetric" matrix 
where formula_14.
Assume that each product of off-diagonal entries is "strictly" positive formula_15 and define a transformation matrix formula_16 by
The similarity transformation formula_18 yields a "symmetric" tridiagonal matrix formula_19 by
Note that formula_21 and formula_19 have the same eigenvalues.
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 (symmetric real) 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.