The inverse of a tridiagonal matrix. (English) Zbl 0980.15004
The author obtains explicit formulae for the elements of the inverse of a general tridiagonal matrix by deriving the explicit solution of a second-order linear nonhomogeneous difference equation with variable coefficients, and then applying the solution to a boundary value problem with zero boundary values. Using the formula for the determinant, he gets an expression for the characteristic polynomial. He also establishes a connection between the matrix inverse and orthogonal polynomials. Moreover he shows how an application of the solution of a second-order linear difference equation to a boundary value problem with periodic boundary conditions can yield the inverse of a cyclic tridiagonal matrix. In the simple case of a tridiagonal or cyclic tridiagonal matrix with constant diagonals, the elements of the inverse can be expressed in terms of the Chebyshev polynomials of the second kind.
Reviewer: Nikolai I.Osetinski (Moskva)
Keywords:
tridiagonal matrix; inverse; second-order linear difference equation; orthogonal polynomials; explicit solutionReferences:
| [1] | Meurant, G., A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 13, 3, 707-728 (1992) · Zbl 0754.65029 |
| [2] | Xu, S.-F., On the Jacobi matrix inverse eigenvalue problem with mixed given data, SIAM J. Matrix Anal. Appl., 17, 3, 632-639 (1996) · Zbl 0856.65032 |
| [3] | Bunse-Gerstner, A.; Byers, R.; Mehrmann, V., A chart of numerical methods for structured eigenvalue problems, SIAM J. Matrix Anal. Appl., 13, 2, 419-453 (1992) · Zbl 0757.65040 |
| [4] | Fischer, C. F.; Usmani, R. A., Properties of some tridiagonal matrices and their application to boundary value problems, SIAM J. Numer. Anal., 6, 1, 127-142 (1969) · Zbl 0176.46802 |
| [5] | Lu, L.; Sun, W., The minimal eigenvalues of a class of block-tridiagonal matrices, IEEE Trans. Inform. Theory, 43, 2, 787-791 (1997) · Zbl 0882.65023 |
| [6] | G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, second ed., Oxford University Press, New York, 1978; G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, second ed., Oxford University Press, New York, 1978 · Zbl 0389.65040 |
| [7] | Mattheij, R. M.M.; Smooke, M. D., Estimates for the inverse of tridiagonal matrices arising in boundary-value problems, Linear Algebra Appl., 73, 33-57 (1986) · Zbl 0592.65012 |
| [8] | G.H. Golub, C.F. Van Loan, Matrix Computations, third ed., Johns Hopkins University Press, Baltimore, MD, 1996; G.H. Golub, C.F. Van Loan, Matrix Computations, third ed., Johns Hopkins University Press, Baltimore, MD, 1996 · Zbl 0865.65009 |
| [9] | Yamamoto, T.; Ikebe, Y., Inversion of band matrices, Linear Algebra Appl., 24, 105-111 (1979) · Zbl 0408.15004 |
| [10] | Barrett, W. W., A theorem on inverses of tridiagonal matrices, Linear Algebra Appl., 27, 211-217 (1979) · Zbl 0414.15004 |
| [11] | Barrett, W. W.; Feinsilver, P. J., Inverses of banded matrices, Linear Algebra Appl., 41, 111-130 (1981) · Zbl 0472.15004 |
| [12] | Romani, F., On the additive structure of the inverses of banded matrices, Linear Algebra Appl., 80, 131-140 (1986) · Zbl 0598.15004 |
| [13] | Nabben, R., Two-sided bounds on the inverses of diagonally dominant tridiagonal matrices, Linear Algebra Appl., 287, 289-305 (1999) · Zbl 0951.15005 |
| [14] | Kelley, W. G.; Peterson, A. C., Difference Equations: An Introduction with Applications (1991), Academic Press: Academic Press San Diego, CA · Zbl 0733.39001 |
| [15] | Elaydi, S. N., An Introduction to Difference Equations (1996), Springer: Springer New York · Zbl 0840.39002 |
| [16] | Mallik, R. K., On the solution of a second order linear homogeneous difference equation with variable coefficients, J. Math. Anal. Appl., 215, 1, 32-47 (1997) · Zbl 0891.39008 |
| [17] | Ikebe, Y., On inverses of Hessenberg matrices, Linear Algebra Appl., 24, 93-97 (1979) · Zbl 0397.15005 |
| [18] | G. Szego, Orthogonal Polynomials, fourth ed., AMS, Providence, RI, 1981; G. Szego, Orthogonal Polynomials, fourth ed., AMS, Providence, RI, 1981 |
| [19] | Elhay, S.; Golub, G. H.; Kautsky, J., Jacobi matrices for sums of weight functions, BIT, 32, 143-166 (1992) · Zbl 0783.65037 |
| [20] | Elhay, S.; Golub, G. H.; Kautsky, J., Updating and downdating of orthogonal polynomials with data fitting applications, SIAM J. Matrix Anal. Appl., 12, 2, 327-353 (1991) · Zbl 0728.65010 |
| [21] | Temperton, C., Algorithms for the solution of cyclic tridiagonal systems, J. Comput. Physics, 19, 317-323 (1975) · Zbl 0319.65024 |
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