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Questions tagged [tridiagonal-matrices]

Relating to all $n\times n$ matrices $(A)$ with the property $A_{i,j}=0$ if $|j-i|>1$

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2answers
20 views

What is meant by tridiagonal linear equation system?

I have to implement the SOR (Successive Over-Relaxation) method, using sparse matrices, to find the solution vector of these linear equations systems (for quite huge matrices): What does that tridiag(...
3
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1answer
68 views

Inverse of a symmetric tridiagonal matrix

I have a symmetric $n\times n$ matrix $\mathbb A$ with entries: $$A_{ij} = (a_i + a_{i-1})\delta_{ij} - a_i\delta_{i,j-1}-a_{j}\delta_{i-1,j}$$ where $a_0,\dots,a_n$ are given positive numbers. Is ...
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0answers
23 views

Eigenpairs of a sparse symmetric block tridiagonal matrix with diagonal blocks on the outer diagonals

I'm looking at methods to find the eigenpairs of symmetric block tridiagonal matrices, with sparse blocks on the main diagonal and diagonal blocks on the outer diagonals. Has any research been done on ...
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1answer
42 views

(Tridiagonal) Inverse of a matrix

Given this $n \times n$ matrix: $$ A= \left(\begin{matrix}a_1&a_1&...&a_1\\a_1&a_2&...&a_2\\\vdots& &\ddots &\vdots\\a_1&a_2&...&a_n\end{matrix}\right) $...
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1answer
31 views

LU composition of a tridiagonal matrix

Given a triadiagonal Matrix A with $$A = \begin{bmatrix} d_1 &e1\\ c_2 & d_2 & e_2 \\ & c_3 & \ddots & \ddots \\ & & \ddots & \ddots & e_{n-1} \\ & & &...
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1answer
39 views

Can this diagonal system be solved by modifying it to the $Ax=B$ form?

I'd like to use the $Ax=B$ form for solving the following system. $$ \left[ \begin{matrix} t_0*d_0 & -t_1*e_0 & 0 & 0 & 0 \\ 0 & t_1*d_1 & -t_2*e_1 & 0 &...
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0answers
11 views

Schur complement in LUP decomposition of a block tridiagonal matrix

Section 2.2 of the article On twisted factorizations of block tridiagonal matrices explains how to do a LUP decomposition of block tridiagonal matrices by showing the process on a 4 blocks by 4 blocks ...
7
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1answer
168 views

Finding the determinant of a tridiagonal matrix

$$\begin{vmatrix}x&1&0&0&⋯\\-n&x-2&2&0&⋯\\0&-(n-1)&x-4&3&⋯\\⋮&⋱&⋱&⋱&⋮\\0&⋯&-2&x-2(n-1)&n\\0&0&⋯&-1&x-2n\...
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0answers
30 views

Can a symmetric positive definite tridiagonal matrix have a zero coefficient on its diagonal?

I'm investigating the Parallel Cyclic Reduction algorithm (Fast Tridiagonal Solvers on the GPU) in the case of positive definite tridiagonal matrices and I'm wondering whether a 0 can appear on the ...
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1answer
42 views

Inverse of symmetric tridiagonal block toeplitz matrix

There is a triagonal block matrix $M$ of form: $$ M = \begin{bmatrix} A & B^T & 0 & 0 & \cdots & 0 & 0 \\ B & A & B^T & 0 & \cdots & 0 & 0 \\ 0 & B ...
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0answers
39 views

Sufficient condition on matrix to tridiagonalize

consider a possibly infinite quadratic matrix $A$. Based on this post: Decomposition into a tridiagonal matrix I am wondering whether there are sufficient conditions aside from $A$ being symmetric ...
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3answers
57 views

Show that the matrix is positive definite

We have the tridiagonal matrix $A=\begin{pmatrix}2 & 1 & \ldots & 0 \\ 1 & 2 & 1 & \ldots \\ \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & 1 & 2\end{...
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1answer
330 views

Creating a Tridiagonal Matrix given 2 Vectors in Matlab

I'm new to Matlab (coming from Python), and I'm trying to figure out the following matrix $A$ given 2 input arrays, $a = [a_0, a_1, ... , a_{n-1}]$, and $b = [b_1, b_2, ... , b_n]$. I tried ...
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2answers
158 views

What real-life examples of a Tridiagonal matrix are there?

I've been looking into the Tridiagonal matrix algorithm. There's theory everywhere but not a lot of real-world applications and examples of tridiagonal matrices. I understand it is a pretty simple and ...
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0answers
81 views

Linear Algebra: tri-diagonal matrix problem.

Let $n\times n$ complex matrices $A=(a_{ij})_{1\leq i,j\leq n}$ and $B=(b_{ij})_{1\leq i,j\leq n}$ be a tri-diagonal matrix whose off-diagonal entries are non-zero and a diagonal matrix, respectively. ...
4
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1answer
87 views

Matrix eigenvalues

Consider the matrix $$A_n=\begin{bmatrix} a & b & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ c & a & b & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & c ...
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0answers
33 views

Is there any specific relationship among the determinant of leading principal submatrices of a tridiagonal matrix?

The following symmetric matrix $A$ is given. Let denote $A_i$ the $i$-th leading principal submatrix of the matrix $A$. $$ A = \begin{bmatrix} a_1 & b_1 \\ b_1 & a_2 & b_2 \\ & b_2 &...
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0answers
108 views

Number of arithmetic operations in Gaussian elimination of a band matrix width w

I have a hard time understanding the number of arithmetic operations required to eliminate a band matrix width $w$ (a matrix with $w$ non-zero diagonals below and above its main diagonal and zeros ...
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9answers
3k views

Is there a fast way to prove a tridiagonal matrix is positive definite?

I' m trying to prove that $$A=\begin{pmatrix} 4 & 2 & 0 & 0 & 0 \\ 2 & 5 & 2 & 0 & 0 \\ 0 & 2 & 5 & 2 & 0 \\ 0 & 0 & 2 & 5 & 2 \\ ...
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1answer
39 views

What is the rank of $B$?

Let $A=(a_{ij})$ be the square matrix of size $2018$ defined by $$ a_{ij} = \begin{cases} 2 & \text{if } i+1=j\\ \frac{1}{3} & \text{if } i =j+1\\ 0 & \text{otherwise}\end{cases}$$ ...
3
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1answer
204 views

Using Cholesky decomposition to solve a system of equaions $A^TAx=b$

I am looking for a way to use $LL^T$ decomposition of a tridiagonal and symmetric positive definite $n$ by $n$ matrix $A$ to solve $$A^TAx=b.$$ In this case, thanks to $A$ being symmetric, the ...
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0answers
52 views

Inversion of a Tridiagonal Matrices and Recurrence equation

I want to find an inverse of an infinite tridiagonal matrix: $ A_{m,n} = (\delta_{m,n} (n-1+\Delta) + s(\sqrt{n} \delta_{m+1,n} + \sqrt{m} \delta_{m,n+1}))_{m,n} $ where $\Delta$ and $s$ are real ...
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2answers
378 views

Generating a random tridiagonal symmetric positive definite matrix

I am looking for a way to generate random tridiagonal symmetric positive definite matrices in Matlab. I need to be able to generate a matrix of specific size (any number between 1-101). Does anybody ...
1
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0answers
103 views

Spectrum of tridiagonal block matrix

I have the following $4n \times 4n$ block tridiagonal matrix: $$ \begin{bmatrix} M_{1} && -M_{2} && 0 && \cdots&&&& &&0 \\ M_{2} && M_{1} &&...
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1answer
128 views

Prove that $Q^{T}TQ$ is symmetric and tridiagonal, where $Q,R$ is $QR$ decomposition of symmetric tridiagonal matrix $T$

Let $T$ be quadratic, invertible, symmetrical tridiagonal matrix and $Q$, $R$ matrices be $QR$ decomposition of $T$. Prove that $Q^{T}TQ$ is also symmetrical and tridiagonal. The symmetric part is ...
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0answers
135 views

Tridiagonal matrix with corners with a lot of symmetry

I really want to find the eigenvalues of this matrix: $$ \left[\begin{array}{ccccccc} a_{0} & 1 & 0 & & 0 & 0 & 1\\ 1 & a_{1} & 1 & \cdots & 0 & 0 & 0\...
1
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0answers
120 views

Why dont we use Given's method to reduce a symmetric matrix to diagonal form rather than tridiagonal form, and why is tridiagonal form desirable?

I've been searching for a while trying to understand why we use Givens method (and householders method) to reduce a symmetric matrix to tri-diagonal form instead of using these methods to reduce a ...
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1answer
51 views

Tridiagonal matrix with charcteristic polynomial is a polynomial $T_k(x)$ given recursively

Let the polynomial sequence $T_k(x)$ as follow: $T_0(x)=1$, $T_1(x)=x$ and $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ for all $n\geq 1$. Show the following: 1) For every $n\geq 2$ find a tridiagonal matrix $...
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0answers
27 views

Largest term in determinant of Banded matrix

I am looking to find the largest term in determinant of Banded matrix, to approximate determinant with it? or is there another approach? Specifically i wants know approximately the determinant of ...
1
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1answer
298 views

Determinant of tridiagonal (banded) matrix

Struggling with homework. I know that this is a banded matrix ( bandwidth = 3) but I don't know how to approach computing the determinant. I tried to compute it with matlab for n= 3,..,7 but I didn't ...
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1answer
430 views

Finding eigenvalues of tridiagonal matrix using MATLAB

$$\begin{bmatrix} -1 & 1 & 0 &\dots &\dots &\dots &0\\ 1 & -2 & 1 & \ddots & & & \vdots\\ 0 & 1 & -2 & \ddots & \ddots & &...
3
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1answer
118 views

Solving a linear system represented by an infinite matrix (tridiagonal)

I want to calculate expected hitting times (or first passage times) in a continuous-time birth-death process. Context Let $Q$ be the instant transition rate matrix and $S = \{0, 1, \ldots\}$ the ...
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0answers
85 views

Positive-definiteness of the summation of a diagonal matrix plus a rank $1$ matrix

I have encountered a problem of determining the condition for matrix $A+A^T$ to be positive definite, where $$A= \operatorname{diag} \{a_1, a_2, \ldots, a_n\} + \mathbb{1}\cdot [b_1, b_2, \ldots, b_n]...
1
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1answer
196 views

Show that a tridiagonal matrix is nonsingular

How would you show that a given tridiagonal matrix is nonsingular? I have a tridiagonal linear system and I would like to show it's nonsingular, which means zero cannot be an eigenvalue. The book I ...
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2answers
142 views

Showing this tridiagonal system of linear equations has a unique solution

Starting with a diagonally dominant tridiagonal $n \times n$ matrix A = \begin{bmatrix} a & b & 0 & 0 & ... & 0 \\ b & a & b & 0 &... & 0 \\ ...
4
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1answer
303 views

Finding eigenvalues in almost tridiagonal matrix

I need to find the eigenvalues of an $n\times n$ symmetric tridiagonal matrix $A$, except it has $1$s on $A_{1n}$ and $A_{n1}$. The diagonal entries are all $4$, while superdiagonal and subdiagonal ...
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0answers
32 views

solve small symmetric triadiagonal eigenvalue problem symbolically

I have a bunch of small ($n<20$) symmetric tridiagonal eigen problems, $$ A x = \lambda x $$ with $A = \text{diag}(b_i, a_i, b_i)$, $i\in\{1,\dots,n\}$, which I need to solve symbolically for ...
2
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1answer
319 views

Inverse of the following symmetric tridiagonal matrix

I have the following matrix: $$ A = \left[ \begin{matrix} -(a_1 + a_2) & a_2 \\ a_2 & -(a_2 + a_3) & a_3\\ & & \ddots & \\ &&a_{n-1}& -(a_{n - 1} + a_{n}...
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0answers
88 views

(revised)-Eigenvalues and eigenvectors of a special tridiagonal matrix (nonsymmetric)?

I need to consider to find the eigenvalues and eigenvectors of the following tridiagonal matrix, \begin{equation} \begin{bmatrix} \gamma & \beta \\ \alpha & \gamma & \beta \\ &...
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1answer
74 views

Eigenvalues of a tridiagonal matrix with boundary conditions

I need to diagonalize(analytically) the following matrix(I really only need the eigenvalues): $$ \begin{matrix} a+e & -i x & 0 & 0 & \cdots & 0\\ i x & ...
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0answers
96 views

Does maximum eigenvalue of the leading principal submatrices of a symmetric tridiagonal matrix always increase?

Given a symmetric tridiagonal matrix in the form of $$\left( {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&{}&{} \\ {{y_1}}&{{x_2}}& \ddots &{} \\ {}& \ddots & \...
3
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1answer
464 views

The algorithm to find an eigenvector of a symmetric tridiagonal matrix associated with a known eigenvalue.

This question is related to my previous question The algorithm to find the largest eigenvalue and one of its eigenvector of a symmetric tridiagonal matrix? The matrix in question is a symmetric ...
3
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0answers
178 views

Eigenvalue, Eigenvector, of a Tridiagonal Symmetric “nearly” Toeplitz Matrix

I am trying to find the eigenvalues/eigenvectors of a NxN tridiagonal symmetric "nearly" Toeplitz matrix, except that a modification on the top left corner \begin{pmatrix} a^2 & -a \\ -a & 1+a^...
2
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1answer
107 views

Partial converse to the diagonal dominance criterion

Diagonal dominance criterion. The tridiagonal symmetric matrix $$\tag{1} A=\begin{bmatrix} a_1 & b_1 & 0& 0& 0 & \ldots & 0 \\ b_1 & a_2 & b_2 & 0 & 0 &...
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0answers
128 views

Tridiagonal With Constant Off-Diagonals

Does anybody know about the eigenvalues of a tridiagonal matrix with constant off-diagonal elements? Such a matrix can be rescaled into $$ \left( \begin{array}{ccccccc} a_1 & 1 & 0& \...
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1answer
81 views

Solving a parabolic PDE

I've been tasked with solving a parabolic PDE (in the form of the Black-Scholes equation), first I need to discretise the problem (which I have done) and then solve to form a vector of approximations ...
2
votes
1answer
51 views

Properties of solutions to system of linear equations with diagnonally dominant, positive (definite) matrix

Consider a system of linear equations $Ax = b$, where \begin{eqnarray} % A & = & \left[ \begin{array}{ccccc} 2 & 1 & & & \\ 1 & 4 & 1 &...
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0answers
115 views

Determinant of a tridiagonal matrix

Let $N \in \mathbb{N}$, and let $A$ be a $(2N+1) \times (2N+1)$ tridiagonal matrix of the following form $$A= \begin{bmatrix} \dots & \dots & \dots & \dots & \dots & \dots & \...
1
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1answer
438 views

Exponential of a symmetric tridiagonal Toeplitz matrix

Let $\alpha$ be a (strictly) positive real number. Consider the following tridiagonal Toeplitz matrix $$ A=\alpha\begin{bmatrix} 0 & 1 & 0 &\cdots & 0\\ 1 & 0 & 1 &\ddots &...
6
votes
1answer
248 views

Eigenvectors of a tridiagonal stochastic matrix

I'm looking for the eigenvectors of this matrix: \begin{equation} \nonumber M = \frac{1}{N} \left( \begin{array}{ccccccccc} 0 & 1 &&&&&&&\\ N & 0 & 2&&...