Editing Tridiagonal matrix (section)

===Inversion===

The [[inverse matrix|inverse]] of a non-singular tridiagonal matrix ''T''

:<math>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}</math>

is given by<!-- (T^{-1})_{ij} = \begin{cases}
(-1)^{j-i}b_i \cdots b_{j-1} \theta_{i-1} \phi_{j+1}/\theta_n & \text{ if } i < j\\
\theta_{i-1} \phi_{j+1}/\theta_n & \text{ if } i = j\\
(-1)^{i-j}c_j \cdots c_{i-1} \theta_{j-1} \phi_{i+1}/\theta_n & \text{ if } i > j\\
\end{cases} Correction!? -->

:<math>(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 < j\\
\theta_{i-1} \phi_{j+1}/\theta_n & \text{ if } i = j\\
(-1)^{i+j}c_j \cdots c_{i-1} \theta_{j-1} \phi_{i+1}/\theta_n & \text{ if } i > j\\
\end{cases}</math>

where the ''θ<sub>i</sub>'' satisfy the recurrence relation

:<math>\theta_i = a_i \theta_{i-1} - b_{i-1}c_{i-1}\theta_{i-2} \qquad i=2,3,\ldots,n</math>

with initial conditions ''θ''<sub>0</sub>&nbsp;=&nbsp;1, ''θ''<sub>1</sub>&nbsp;=&nbsp;''a''<sub>1</sub> and the ''ϕ''<sub>''i''</sub> satisfy

:<math>\phi_i = a_i \phi_{i+1} - b_i c_i \phi_{i+2} \qquad i=n-1,\ldots,1</math>

with initial conditions ''ϕ''<sub>''n''+1</sub>&nbsp;=&nbsp;1 and ''ϕ''<sub>''n''</sub>&nbsp;=&nbsp;''a<sub>n</sub>''.<ref>{{Cite journal | last1 = Da Fonseca | first1 = C. M. | doi = 10.1016/j.cam.2005.08.047 | title = On the eigenvalues of some tridiagonal matrices | journal = Journal of Computational and Applied Mathematics | volume = 200 | pages = 283–286 | year = 2007 | doi-access = free }}</ref><ref>{{Cite journal | last1 = Usmani | first1 = R. A. | doi = 10.1016/0024-3795(94)90414-6 | title = Inversion of a tridiagonal jacobi matrix | journal = Linear Algebra and Its Applications | volume = 212-213 | pages = 413–414 | year = 1994 | doi-access = free }}</ref>

Closed form solutions can be computed for special cases such as [[symmetric matrix|symmetric matrices]] with all diagonal and off-diagonal elements equal<ref>{{Cite journal | last1 = Hu | first1 = G. Y. | last2 = O'Connell | first2 = R. F. | doi = 10.1088/0305-4470/29/7/020 | title = Analytical inversion of symmetric tridiagonal matrices | journal = Journal of Physics A: Mathematical and General | volume = 29 | issue = 7 | pages = 1511 | year = 1996 | bibcode = 1996JPhA...29.1511H }}</ref> or [[Toeplitz matrices]]<ref>{{Cite journal | last1 = Huang | first1 = Y. | last2 = McColl | first2 = W. F. | doi = 10.1088/0305-4470/30/22/026 | title = Analytical inversion of general tridiagonal matrices | journal = Journal of Physics A: Mathematical and General | volume = 30 | issue = 22 | pages = 7919 | year = 1997 | bibcode = 1997JPhA...30.7919H }}</ref> and for the general case as well.<ref>{{Cite journal | last1 = Mallik | first1 = R. K. | doi = 10.1016/S0024-3795(00)00262-7 | title = The inverse of a tridiagonal matrix | journal = Linear Algebra and Its Applications | volume = 325 | pages = 109–139 | year = 2001 | issue = 1–3 | doi-access = free }}</ref><ref>{{Cite journal | last1 = Kılıç | first1 = E. | doi = 10.1016/j.amc.2007.07.046 | title = Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions | journal = Applied Mathematics and Computation | volume = 197 | pages = 345–357 | year = 2008 }}</ref>

In general, the inverse of a tridiagonal matrix is a [[semiseparable matrix]] and vice versa.<ref name="VandebrilBarel2008">{{cite book|author1=Raf Vandebril|author2=Marc Van Barel|author3=Nicola Mastronardi|title=Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems|year=2008|publisher=JHU Press|isbn=978-0-8018-8714-7|at=Theorem 1.38, p. 41}}</ref>  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<ref name="Meurant1992">{{cite journal |last1=Meurant |first1=Gerard |title=A review on the inverse of symmetric tridiagonal and block tridiagonal matrices |journal=SIAM Journal on Matrix Analysis and Applications |date=1992 |volume=13 |issue=3 |pages=707–728 |doi=10.1137/0613045 |url=https://doi.org/10.1137/0613045|url-access=subscription }}</ref><ref>{{cite journal |last1=Bossu |first1=Sebastien |title=Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices |journal=Linear Algebra and Its Applications |date=2024 |volume=699 |pages=129–158 |doi=10.1016/j.laa.2024.06.018 |url=https://authors.elsevier.com/a/1jOTP5YnCtZEc|arxiv=2304.06100 }}</ref>

<math>\begin{pmatrix}
        \alpha_1 & -\beta_1 \\
        -\beta_1  & \alpha_2 & -\beta_2 \\
        & \ddots & \ddots & \ddots & \\
& & \ddots & \ddots & -\beta_{n-1} \\
& & & -\beta_{n-1} & \alpha_n
        \end{pmatrix}^{-1} = 
\begin{pmatrix}
        a_1 b_1 & a_1 b_2 & \cdots & a_1 b_n \\
        a_1 b_2 & a_2 b_2 & \cdots & a_2 b_n \\
        \vdots  & \vdots & \ddots & \vdots \\
        a_1 b_n & a_2 b_n & \cdots & a_n b_n
    \end{pmatrix}
= \left( a_{\min(i,j)} b_{\max(i,j)} \right)
</math>

where <math>\begin{cases} \displaystyle a_i = \frac{\beta_{i} \cdots \beta_{n-1}}{\delta_i \cdots \delta_n\,b_n}
\\ \displaystyle
b_i = \frac{\beta_1 \cdots \beta_{i-1}}{d_1 \cdots d_i}\end{cases}</math>
with <math>\begin{cases}
        d_n = \alpha_n,\quad d_{i-1} = \alpha_{i-1} - \frac{\beta_{i-1}^2}{d_{i}}, & i = n, n-1, \cdots, 2,
        \\
        \delta_1 = \alpha_1, \quad
        \delta_{i+1} = \alpha_{i+1} - \frac{\beta_{i}^2}{\delta_{i}}, & i = 1, 2, \cdots, n-1.
    \end{cases}
</math>