Questions tagged [tridiagonal-matrices]
Relating to all $n\times n$ matrices $(A)$ with the property $A_{i,j}=0$ if $|j-i|>1$
0
votes
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
votes
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 ...
1
vote
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 ...
1
vote
1answer
42 views
(Tridiagonal) Inverse of a matrix
Given this $n \times n$ matrix:
$$ A= \left(\begin{matrix}a_1&a_1&...&a_1\\a_1&a_2&...&a_2\\\vdots& &\ddots &\vdots\\a_1&a_2&...&a_n\end{matrix}\right) $...
0
votes
1answer
31 views
LU composition of a tridiagonal matrix
Given a triadiagonal Matrix A with
$$A = \begin{bmatrix}
d_1 &e1\\
c_2 & d_2 & e_2 \\
& c_3 & \ddots & \ddots \\
& & \ddots & \ddots & e_{n-1} \\
& & &...
0
votes
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.
$$
\left[
\begin{matrix}
t_0*d_0 & -t_1*e_0 & 0 & 0 & 0 \\
0 & t_1*d_1 & -t_2*e_1 & 0 &...
0
votes
0answers
11 views
Schur complement in LUP decomposition of a block tridiagonal matrix
Section 2.2 of the article On twisted factorizations of block tridiagonal matrices explains how to do a LUP decomposition of block tridiagonal matrices by showing the process on a 4 blocks by 4 blocks ...
7
votes
1answer
168 views
Finding the determinant of a tridiagonal matrix
$$\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\...
0
votes
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 ...
0
votes
1answer
42 views
Inverse of symmetric tridiagonal block toeplitz matrix
There is a triagonal block matrix $M$ of form:
$$
M = \begin{bmatrix}
A & B^T & 0 & 0 & \cdots & 0 & 0 \\
B & A & B^T & 0 & \cdots & 0 & 0 \\
0 & B ...
0
votes
0answers
39 views
Sufficient condition on matrix to tridiagonalize
consider a possibly infinite quadratic matrix $A$. Based on this post: Decomposition into a tridiagonal matrix I am wondering whether there are sufficient conditions aside from $A$ being symmetric ...
0
votes
3answers
57 views
Show that the matrix is positive definite
We have the tridiagonal matrix $A=\begin{pmatrix}2 & 1 & \ldots & 0 \\ 1 & 2 & 1 & \ldots \\ \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & 1 & 2\end{...
0
votes
1answer
330 views
Creating a Tridiagonal Matrix given 2 Vectors in Matlab
I'm new to Matlab (coming from Python), and I'm trying to figure out the following matrix $A$ given 2 input arrays, $a = [a_0, a_1, ... , a_{n-1}]$, and $b = [b_1, b_2, ... , b_n]$.
I tried ...
0
votes
2answers
158 views
What real-life examples of a Tridiagonal matrix are there?
I've been looking into the Tridiagonal matrix algorithm. There's theory everywhere but not a lot of real-world applications and examples of tridiagonal matrices. I understand it is a pretty simple and ...
1
vote
0answers
81 views
Linear Algebra: tri-diagonal matrix problem.
Let $n\times n$ complex matrices $A=(a_{ij})_{1\leq i,j\leq n}$ and $B=(b_{ij})_{1\leq i,j\leq n}$ be a tri-diagonal matrix whose off-diagonal entries are non-zero and a diagonal matrix, respectively. ...
4
votes
1answer
87 views
Matrix eigenvalues
Consider the matrix $$A_n=\begin{bmatrix}
a & b & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\
c & a & b & 0 & 0 & \dots & 0 & 0 & 0 \\
0 & c ...
2
votes
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 $A$.
$$
A = \begin{bmatrix}
a_1 & b_1 \\
b_1 & a_2 & b_2 \\
& b_2 &...
0
votes
0answers
108 views
Number of arithmetic operations in Gaussian elimination of a band matrix width w
I have a hard time understanding the number of arithmetic operations required to eliminate a band matrix width $w$ (a matrix with $w$ non-zero diagonals below and above its main diagonal and zeros ...
29
votes
9answers
3k views
Is there a fast way to prove a tridiagonal matrix is positive definite?
I' m trying to prove that
$$A=\begin{pmatrix}
4 & 2 & 0 & 0 & 0 \\
2 & 5 & 2 & 0 & 0 \\
0 & 2 & 5 & 2 & 0 \\
0 & 0 & 2 & 5 & 2 \\
...
4
votes
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
votes
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 ...
1
vote
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 ...
1
vote
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
vote
0answers
103 views
Spectrum of tridiagonal block matrix
I have the following $4n \times 4n$ block tridiagonal matrix:
$$
\begin{bmatrix}
M_{1} && -M_{2} && 0 && \cdots&&&& &&0 \\
M_{2} && M_{1} &&...
0
votes
1answer
128 views
Prove that $Q^{T}TQ$ is symmetric and tridiagonal, where $Q,R$ is $QR$ decomposition of symmetric tridiagonal matrix $T$
Let $T$ be quadratic, invertible, symmetrical tridiagonal matrix and $Q$, $R$ matrices be $QR$ decomposition of $T$. Prove that $Q^{T}TQ$ is also symmetrical and tridiagonal.
The symmetric part is ...
5
votes
0answers
135 views
Tridiagonal matrix with corners with a lot of symmetry
I really want to find the eigenvalues of this matrix:
$$
\left[\begin{array}{ccccccc}
a_{0} & 1 & 0 & & 0 & 0 & 1\\
1 & a_{1} & 1 & \cdots & 0 & 0 & 0\...
1
vote
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 ...
0
votes
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 $...
1
vote
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
vote
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 ...
1
vote
1answer
430 views
Finding eigenvalues of tridiagonal matrix using MATLAB
$$\begin{bmatrix}
-1 & 1 & 0 &\dots &\dots &\dots &0\\
1 & -2 & 1 & \ddots & & & \vdots\\
0 & 1 & -2 & \ddots & \ddots & &...
3
votes
1answer
118 views
Solving a linear system represented by an infinite matrix (tridiagonal)
I want to calculate expected hitting times (or first passage times) in a continuous-time birth-death process.
Context
Let $Q$ be the instant transition rate matrix and $S = \{0, 1, \ldots\}$ the ...
1
vote
0answers
85 views
Positive-definiteness of the summation of a diagonal matrix plus a rank $1$ matrix
I have encountered a problem of determining the condition for matrix $A+A^T$ to be positive definite, where
$$A= \operatorname{diag} \{a_1, a_2, \ldots, a_n\} + \mathbb{1}\cdot [b_1, b_2, \ldots, b_n]...
1
vote
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 ...
0
votes
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
votes
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 ...
0
votes
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
votes
1answer
319 views
Inverse of the following symmetric tridiagonal matrix
I have the following matrix:
$$
A = \left[
\begin{matrix}
-(a_1 + a_2) & a_2 \\
a_2 & -(a_2 + a_3) & a_3\\
& & \ddots & \\
&&a_{n-1}& -(a_{n - 1} + a_{n}...
2
votes
0answers
88 views
(revised)-Eigenvalues and eigenvectors of a special tridiagonal matrix (nonsymmetric)?
I need to consider to find the eigenvalues and eigenvectors of the following tridiagonal matrix,
\begin{equation}
\begin{bmatrix}
\gamma & \beta \\
\alpha & \gamma & \beta \\
&...
0
votes
1answer
74 views
Eigenvalues of a tridiagonal matrix with boundary conditions
I need to diagonalize(analytically) the following matrix(I really only need the eigenvalues):
$$
\begin{matrix}
a+e & -i x & 0 & 0 & \cdots & 0\\
i x & ...
0
votes
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
$$\left( {\begin{array}{*{20}{c}}
{{x_1}}&{{y_1}}&{}&{} \\
{{y_1}}&{{x_2}}& \ddots &{} \\
{}& \ddots & \...
3
votes
1answer
464 views
The algorithm to find an eigenvector of a symmetric tridiagonal matrix associated with a known eigenvalue.
This question is related to my previous question The algorithm to find the largest eigenvalue and one of its eigenvector of a symmetric tridiagonal matrix? The matrix in question is a symmetric ...
3
votes
0answers
178 views
Eigenvalue, Eigenvector, of a Tridiagonal Symmetric “nearly” Toeplitz Matrix
I am trying to find the eigenvalues/eigenvectors of a NxN tridiagonal symmetric "nearly" Toeplitz matrix, except that a modification on the top left corner
\begin{pmatrix}
a^2 & -a \\
-a & 1+a^...
2
votes
1answer
107 views
Partial converse to the diagonal dominance criterion
Diagonal dominance criterion.
The tridiagonal symmetric matrix
$$\tag{1} A=\begin{bmatrix}
a_1 & b_1 & 0& 0& 0 & \ldots & 0 \\
b_1 & a_2 & b_2 & 0 & 0 &...
0
votes
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
$$
\left( \begin{array}{ccccccc}
a_1 & 1 & 0& \...
1
vote
1answer
81 views
Solving a parabolic PDE
I've been tasked with solving a parabolic PDE (in the form of the Black-Scholes equation), first I need to discretise the problem (which I have done) and then solve to form a vector of approximations ...
2
votes
1answer
51 views
Properties of solutions to system of linear equations with diagnonally dominant, positive (definite) matrix
Consider a system of linear equations $Ax = b$, where
\begin{eqnarray}
%
A & = & \left[ \begin{array}{ccccc}
2 & 1 & & & \\
1 & 4 & 1 &...
1
vote
0answers
115 views
Determinant of a tridiagonal matrix
Let $N \in \mathbb{N}$, and let $A$ be a $(2N+1) \times (2N+1)$ tridiagonal matrix of the following form
$$A=
\begin{bmatrix}
\dots & \dots & \dots & \dots & \dots & \dots & \...
1
vote
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&&...
