Tridiagonal matrix

From formulasearchengine
Jump to navigation Jump to search

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:

The determinant of a tridiagonal matrix is given by the continuant of its elements.[1]

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


A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix.[2] 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 ak,k+1 ak+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 ak,k+1 ak+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.[3]

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.


{{#invoke:main|main}} The determinant of a tridiagonal matrix A of order n can be computed from a three-term recurrence relation.[4] Write f1 = |a1| = a1 and

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

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.


The inverse of a non-singular tridiagonal matrix T

is given by

where the θi satisfy the recurrence relation

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

with initial conditions ϕn+1 = 1 and ϕn = an.[5][6]

Closed form solutions can be computed for special cases such as symmetric matrices with all off-diagonal elements equal[7] or Toeplitz matrices[8] and for the general case as well.[9][10]

Solution of linear system

{{#invoke:main|main}} A system of equations A x = b for  can be solved by an efficient form of Gaussian elimination when A is tridiagonal called tridiagonal matrix algorithm, requiring O(n) operations.[11]


When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, namely , for [12][13]

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


  1. {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}
  3. Horn & Johnson, page 174
  4. Template:Cite doi
  5. Template:Cite doi
  6. Template:Cite doi
  7. Template:Cite doi
  8. Template:Cite doi
  9. Template:Cite doi
  10. Template:Cite doi
  11. {{#invoke:citation/CS1|citation |CitationClass=book }}
  12. Template:Cite doi
  13. This can also be written as because , as is done in: Template:Cite doi

External links

|CitationClass=journal }}