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Tridiagonal matrix

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In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main diagonal). For example, the following matrix is tridiagonal:

(1400341002340013).

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.

Properties

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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.

Determinant

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The determinant of a tridiagonal matrix A of order n can be computed from a three-term recurrence relation.[4] Write f1 = |a1| = a1 (i.e., f1 is the determinant of the 1 by 1 matrix consisting only of a1), and let

fn=|a1b1c1a2b2c2⋱⋱⋱⋱bn−1cn−1an|.

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

fn=anfn−1−cn−1bn−1fn−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

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The inverse of a non-singular tridiagonal matrix T

T=(a1b1c1a2b2c2⋱⋱⋱⋱bn−1cn−1an)

is given by

(T−1)ij={(−1)i+jbi⋯bj−1θi−1ϕj+1/θn if i<jθi−1ϕj+1/θn if i=j(−1)i+jcj⋯ci−1θj−1ϕi+1/θn if i>j

where the θi satisfy the recurrence relation

θi=aiθi−1−bi−1ci−1θi−2i=2,3,…,n

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

ϕi=aiϕi+1−biciϕi+2i=n−1,…,1

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 diagonal and off-diagonal elements equal[7] or Toeplitz matrices[8] and for the general case as well.[9][10]

In general, the inverse of a tridiagonal matrix is a semiseparable matrix and vice versa.[11] The inverse of a symmetric tridiagonal matrix can be written as a single-pair matrix (a.k.a. generator-representable semiseparable matrix) of the form[12][13]

(α1−β1−β1α2−β2⋱⋱⋱⋱⋱−βn−1−βn−1αn)−1=(a1b1a1b2⋯a1bna1b2a2b2⋯a2bn⋮⋮⋱⋮a1bna2bn⋯anbn)=(amin⁡(i,j)bmax⁡(i,j))

where {ai=βi⋯βn−1δi⋯δnbnbi=β1⋯βi−1d1⋯di with {dn=αn,di−1=αi−1−βi−12di,i=n,n−1,⋯,2,δ1=α1,δi+1=αi+1−βi2δi,i=1,2,⋯,n−1.

Solution of linear system

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A system of equations Ax = b for b∈ℝn can be solved by an efficient form of Gaussian elimination when A is tridiagonal called tridiagonal matrix algorithm, requiring O(n) operations.[14]

Eigenvalues

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When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, namely:[15][16]

a−2bccos⁡(kπn+1),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.[17] Numerous methods exist for the numerical computation of the eigenvalues of a real symmetric tridiagonal matrix to arbitrary finite precision, typically requiring O(n2) operations for a matrix of size n×n, although fast algorithms exist which (without parallel computation) require only O(nlog⁡n).[18]

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.[19]

Similarity to symmetric tridiagonal matrix

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For unsymmetric or nonsymmetric tridiagonal matrices one can compute the eigendecomposition using a similarity transformation. Given a real tridiagonal, nonsymmetric matrix

T=(a1b1c1a2b2c2⋱⋱⋱⋱bn−1cn−1an)

where bi≠ci. Assume that each product of off-diagonal entries is strictly positive bici>0 and define a transformation matrix D by[20]

D:=diag⁡(δ1,…,δn)forδi:={1,i=1ci−1…c1bi−1…b1,i=2,…,n.

The similarity transformation D−1TD yields a symmetric tridiagonal matrix J by:[21][20]

J:=D−1TD=(a1sgn⁡b1b1c1sgn⁡b1b1c1a2sgn⁡b2b2c2sgn⁡b2b2c2⋱⋱⋱⋱sgn⁡bn−1bn−1cn−1sgn⁡bn−1bn−1cn−1an).

Note that T and J have the same eigenvalues.

Computer programming

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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.[22]

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.

Applications

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The discretization in space of the one-dimensional diffusion or heat equation

∂u(t,x)∂t=α∂2u(t,x)∂x2

using second order central finite differences with Dirichlet boundary condition results in

(∂u1(t)∂t∂u2(t)∂t⋮∂uN(t)∂t)=αΔx2(−211−211⋱⋱⋱⋱11−2)(u1(t)u2(t)⋮uN(t))

with discretization constant Δx. The matrix is tridiagonal with ai=−2 and bi=ci=1, and so is also a Toeplitz matrix.

When considering instead Neumann boundary condition, the second derivative operator is approximated by the matrix (−111−211⋱⋱⋱⋱11−1)which is tridiagonal but not Toeplitz.

Note that when considering Periodic boundary conditions, the discretization of the second derivative operator gives rise to a matrix that is not tridiagonal, even if the same stencil is used.

See also

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Notes

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  1. ↑ Thomas Muir (1960). A treatise on the theory of determinants. Dover Publications. pp. 516–525.
  2. ↑ Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis. Cambridge University Press. p. 28. ISBN 0521386322.
  3. ↑ Horn & Johnson, page 174
  4. ↑ El-Mikkawy, M. E. A. (2004). "On the inverse of a general tridiagonal matrix". Applied Mathematics and Computation. 150 (3): 669–679. doi:10.1016/S0096-3003(03)00298-4.
  5. ↑ Da Fonseca, C. M. (2007). "On the eigenvalues of some tridiagonal matrices". Journal of Computational and Applied Mathematics. 200: 283–286. doi:10.1016/j.cam.2005.08.047. hdl:10316/4608.
  6. ↑ Usmani, R. A. (1994). "Inversion of a tridiagonal jacobi matrix". Linear Algebra and Its Applications. 212–213: 413–414. doi:10.1016/0024-3795(94)90414-6.
  7. ↑ Hu, G. Y.; O'Connell, R. F. (1996). "Analytical inversion of symmetric tridiagonal matrices". Journal of Physics A: Mathematical and General. 29 (7): 1511. Bibcode:1996JPhA...29.1511H. doi:10.1088/0305-4470/29/7/020.
  8. ↑ Huang, Y.; McColl, W. F. (1997). "Analytical inversion of general tridiagonal matrices". Journal of Physics A: Mathematical and General. 30 (22): 7919. Bibcode:1997JPhA...30.7919H. doi:10.1088/0305-4470/30/22/026.
  9. ↑ Mallik, R. K. (2001). "The inverse of a tridiagonal matrix". Linear Algebra and Its Applications. 325 (1–3): 109–139. doi:10.1016/S0024-3795(00)00262-7.
  10. ↑ Kılıç, E. (2008). "Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions". Applied Mathematics and Computation. 197: 345–357. doi:10.1016/j.amc.2007.07.046.
  11. ↑ Raf Vandebril; Marc Van Barel; Nicola Mastronardi (2008). Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems. JHU Press. Theorem 1.38, p. 41. ISBN 978-0-8018-8714-7.
  12. ↑ Meurant, Gerard (1992). "A review on the inverse of symmetric tridiagonal and block tridiagonal matrices". SIAM Journal on Matrix Analysis and Applications. 13 (3): 707–728. doi:10.1137/0613045.
  13. ↑ Bossu, Sebastien (2024). "Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices". Linear Algebra and Its Applications. 699: 129–158. arXiv:2304.06100. doi:10.1016/j.laa.2024.06.018.
  14. ↑ Golub, Gene H.; Van Loan, Charles F. (1996). Matrix Computations (3rd ed.). The Johns Hopkins University Press. ISBN 0-8018-5414-8.
  15. ↑ Noschese, S.; Pasquini, L.; Reichel, L. (2013). "Tridiagonal Toeplitz matrices: Properties and novel applications". Numerical Linear Algebra with Applications. 20 (2): 302. doi:10.1002/nla.1811.
  16. ↑ This can also be written as a+2bccos⁡(kπ/(n+1)) because cos⁡(x)=−cos⁡(π−x), as is done in: Kulkarni, D.; Schmidt, D.; Tsui, S. K. (1999). "Eigenvalues of tridiagonal pseudo-Toeplitz matrices" (PDF). Linear Algebra and Its Applications. 297 (1–3): 63–80. doi:10.1016/S0024-3795(99)00114-7.
  17. ↑ Parlett, B.N. (1997) [1980]. The Symmetric Eigenvalue Problem. Classics in applied mathematics. Vol. 20. SIAM. ISBN 0-89871-402-8. OCLC 228147822.
  18. ↑ Coakley, E.S.; Rokhlin, V. (2012). "A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices". Applied and Computational Harmonic Analysis. 34 (3): 379–414. doi:10.1016/j.acha.2012.06.003.
  19. ↑ Dhillon, Inderjit Singh (1997). A New O(n2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem (PDF) (PhD). University of California, Berkeley. p. 8. CSD-97-971, ADA637073.
  20. 1 2 Kreer, M. (1994). "Analytic birth-death processes: a Hilbert space approach". Stochastic Processes and Their Applications. 49 (1): 65–74. doi:10.1016/0304-4149(94)90112-0.
  21. ↑ Meurant, Gérard (October 2008). "Tridiagonal matrices" (PDF) – via Institute for Computational Mathematics, Hong Kong Baptist University.
  22. ↑ Eidelman, Yuli; Gohberg, Israel; Gemignani, Luca (2007-01-01). "On the fast reduction of a quasiseparable matrix to Hessenberg and tridiagonal forms". Linear Algebra and Its Applications. 420 (1): 86–101. doi:10.1016/j.laa.2006.06.028. ISSN 0024-3795.
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