Questions tagged [tridiagonal-matrices]
Relating to all $n\times n$ matrices $(A)$ with the property $a_{i,j}=0$ if $|j-i|>1$
119
questions
1
vote
1answer
47 views
$\|\cdot\|_2$ norm of tridiagonal matrix
Let $T\in M_{n}(\mathbb{R})$ be a tridiagonal matrix. What can we say about operator norm $\|T\|_2$?
I'm asking this question because we know that if $T$ were only diagonal, then $\|T\|_2$ is the ...
1
vote
0answers
28 views
Eigenvalues of a particular tridiagonal matrix with a variation on one of its diagonal entries
I am trying to obtain the eigenvalues of this particular $n\times n$ tridiagonal matrix
\begin{eqnarray}
A & = &
\begin{bmatrix}
1+a^{2} & -a & 0 & \ldots & 0 & 0 \\
-a ...
1
vote
0answers
15 views
Eigenvalues of symmetric tridiagonal matrices with identical off diagonal elements
Is there a simple analytical solution to obtain eigenvalues (and eigenvectors) for this type of tridiagonal matrices ? ( Off diagonal elements are identical and the matrix is symmetric)
$ \begin{...
1
vote
0answers
27 views
Signature of a tridiagonal matrix
Let $A$ be a tridiagonal matrix of the form
$$A=
\begin{pmatrix}
a_1 & 1 \\
1 & a_2 & 1 \\
& 1 & \ddots & \ddots \\
& & \ddots & \ddots & 1 \\
& & & ...
0
votes
1answer
31 views
Why does tridiagonal matrix reduce noise?
Let
$$B = \begin{pmatrix}
1/3 & 1/3 & 0 & 0 & 0 & \dots & 0 & 0 \\
1/3 & 1/3 & 1/3 & 0 & 0 & \dots & 0 & 0 \\
0 & 1/3 & 1/3 & 1/3 &...
1
vote
1answer
11 views
Optimizing the inverse of a symmetric matrix with 3 non consecutive diagonal terms
Let $A$ be a $200x200$ matrix that is symmetric and has 3 diagonal terms. It is like a tridiagonal matrix but the diagonals are non-consecutive. Or like a banded matrix with bandwidth 100 but with ...
0
votes
1answer
15 views
denseness of tridiagonal positive semidefinitness in tridiagonal positive definiteness
I know that pd (positive definite) matrices are dense in psd matrices. Let $A$ be TPSD (tridiagonal positive semi definite) with all entries positive. Does there exist a sequence of TPD matrices with ...
0
votes
0answers
21 views
Tridiagonal SPD matrix decomposition
I solved the first part of the question by the formula of matrix multiplication and contradiction. However, the second and third part seems a bit confusing for me.
I tried to combine Cholesky ...
0
votes
0answers
12 views
s'th main diagonal of the inverse of a symmetric tridiagonal matrix
Given a symmetric tridiagonal (diagonally dominant) real matrix A,
$$
A = \begin{pmatrix}a_1 & -b_1 & 0 & 0 &\ldots \\ -b_1 & a_2 & -b_2 & 0 &\ldots \\ 0 & -b_2 &...
0
votes
0answers
46 views
Block Tridiagonal Matrix Eigenvectors and Eigenvalues
I want to find the eigenvectors and eigenvalues of the following $2L \times 2L$ (assume $L$ is even) block tridiagonal matrix,
$$
\begin{pmatrix}
R_{\phantom{1}} & R_{1} & 0 & 0 & 0 &...
1
vote
0answers
35 views
Why does Relative Robust Representations (RRR) of Matrices determine eigenvalues to high relative accuracy?
I am trying to understand the MRRR algorithm for finding eigenpairs. As part of this I am reading Parletts and Dhillons paper on "Relatively robust representations of symmetric tridiagonals". I am ...
0
votes
0answers
38 views
Determinant of a tridiagonal block matrix
I am looking to find the determinant of the following $N^2\times N^2$
matrix $$M =
\begin{pmatrix}
P_N & D_N^{-1} & & D_N \\
D_N & P_N & \ddots & \\
& \ddots &...
0
votes
1answer
37 views
Numerical methods, show that wave equation expression is constant
$\newcommand{\pd}[2]{\frac{\partial#1}{\partial#2}}
\newcommand{\IR}{\Bbb R}\newcommand{\dd}{\mathrm{d}}$
The wave equation:
$$
\pd{^2u}{t^2}(x,t)=c^2\pd{^2u}{x^2}(x,t),~~ t>0,~x\in(0,1)\tag4
$$
...
1
vote
1answer
51 views
Show that $\det A_n = \sum_{i=0}^{n}a^{2i}$
Let $\mathbb{F}$ be a field and $a\in\mathbb{F}$. For $n \geq 1$ let
$$A_n=\begin{pmatrix} 1+a^2 & a & 0 & 0 & \ldots & 0 \\ a & 1+a^2 & a & 0 & \ldots & 0 \\...
1
vote
0answers
31 views
What are some basic methods to solve a recursive function of the shape $f_n = \alpha f_{n-1} + \beta f_{n-2}$ where $\alpha, \beta \in\mathbb{R}$ [duplicate]
I want to understand a little bit more about determining the determinant of tridiagonal matrices. These determinants are (mostly) of the form
$$f_n = \alpha f_{n-1} + \beta f_{n-2}$$ where $\alpha, \...
4
votes
2answers
81 views
How to prove this formula for the determinant of a $4 \times 4$ tridiagonal matrix?
This following is a problem from B. S. Grewal's Higher Engineering Mathematics.
Show
$$\begin{vmatrix} 2\cos(\theta) & 1 & 0 & 0 \\ 1 & 2 \cos(\theta) & 1
& 0 \\ 0 &...
2
votes
1answer
92 views
Inverse of a symmetric block tridiagonal matrix
I am aware of existent discussion on the inverse of a block tridiagonal matrix on this website (for example, How to invert a block tridiagonal matrix?) and I have been googling articles about this ...
0
votes
1answer
41 views
How to write block matrices on diagonal in nice form?
Let $Y$ be real $2\times 2$ matrix
\begin{equation*}
Y = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\end{equation*}
and $Z$ is block matrix constructed as depicted on the picture bellow
...
1
vote
0answers
32 views
Eigenvalue and Eigenvector of specific tridiagonal matrix
I would like to know if exist the analytical solution for the eingenvalue
to the following tridiagonal matrix:
$A_{i,i}=i$ for $i=1,2,...,N$,
$A_{i,i+1}=A_{i+1,i}=1$
$$
\left(\begin{matrix}
...
0
votes
1answer
95 views
Eigenvalues for a block matrix with Toeplitz tridiagonal sub-matrix
Given a matrix $M \in \mathbb{R}^{(2N, 2N)}$ for some $N \in \mathbb{Z}, N > 2$
$$M = \begin{pmatrix}\textbf{0}&I\\A&\textbf{0}\end{pmatrix},$$
where $\textbf{0} \in \mathbb{R}^{(...
2
votes
1answer
59 views
Inverse of tridiagonal Toeplitz matrix has no zero entries
The inverse of the symmetric tridiagonal matrix (Toeplitz)
$$
t_{ij}=\begin{align}
\begin{cases}
-2 &\quad \text{if} \,\, i=j \\
1 &\quad\text{if} \,\, \vert i-j\vert = 1
\...
0
votes
0answers
38 views
Lanczos on structured matrix
Suppose $T ā \mathbb R^{n\times n}$ is tridiagonal and symmetric and that $vā\mathbb R^n$. Show how the Lanczos process can be used (in principle) to compute an orthogonal $Q ā \mathbb R^{n\times n}$ ...
0
votes
1answer
106 views
Eigenvalues of a tridiagonal Toeplitz Matrix and its spectral radius
Let $A$ be the $n \times n$ tridiagonal Toeplitz matrix of the form
$$A = \left[ \begin{array}{cccccc}
2 & -1 & 0 & \dots & \dots & 0 \\
-1 & 2 & -1 & 0 & \...
1
vote
0answers
50 views
Proving a statement about the determinant of tridiagonal matrix [duplicate]
The statement of the question is given below:
In order to understand the meaning of tridiagonal, I tried to calculate the $4 \times 4$ matrix of the above description and I get the following ...
-2
votes
1answer
38 views
Solve the matrices equation AX=B (tridiagonal matrices)
Solve this equation AX=B
\begin{align}
\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
0 & 1 & 1 & \ddots & 1 \\
\vdots & \ddots & \ddots & \ddots ...
3
votes
2answers
125 views
Find the determinant whose result is $(x-n)^{n+1}$
Find the determinant
$$
\left|\begin{array}{cccccc}{x} & {1} & {} & {} & {} & {} \\ {-n} & {x-2} & {2} & {} & {} & {} \\ {} & {-(n-1)} & {x-4} & {\...
0
votes
0answers
45 views
On a recurrence relation with non-constant coefficients
I am studying a real symmetric tridiagonal matrix $J_{N+1}$ (all off-diagonal elements non-zero) of dimension $N+1$, and I would like to solve the eigenvalue problem. The point is that the ...
0
votes
1answer
44 views
Determinant of a tridiagonal matrix using recurrence formula
I'd like to get the determinant $|P_n|$ with :
$$P_n=
\begin{bmatrix}
p+q & q & 0 & \dots & 0 & 0 \\
p & p+q & \ddots & \ddots & \vdots &\vdots \\
0 & p &...
1
vote
1answer
98 views
QR Algorithm and tridiagonal matrix
Consider $A: n \times n$ a symmetric matrix.
a) Explain the main steps to perform on Orthogonal transformations so as to
to obtain a matrix $T$, tridiagonal, orthogonally similar to $A$.
b) ...
1
vote
0answers
37 views
Number of operations needed to solve tridiagonal linear system
I was doing this question. I just checked some posts related to it, but didn't find anything suitable.
1
vote
2answers
101 views
Positivity of eigenvalues of tridiagonal, almost-Toeplitz matrix
How can one show that the following tridiagonal matrix
$$M_n= \begin{pmatrix}
-1&3&0&\dots&\dots&\dots&0\\
3&2&-1&0&&&\vdots\\
0&-1&2&-...
1
vote
0answers
82 views
Convolution of element of matrix exponential
I need to take a $d$-dimensional convolution of the last element of a matrix exponential, but am having trouble. Some specifics:
For a positive integer $g$, I have the $g \times g$ matrix
$$
Q = ...
0
votes
2answers
33 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(...
0
votes
0answers
162 views
Complexity of QR-algorithm on a symmetric tridiagonal matrix
Why does the QR algorithm (for calculating eigenvalues) only require O(m) calculations per step when performed on a symmetric tridiagonal matrix?
3
votes
1answer
147 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
51 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
70 views
(Tridiagonal) Inverse of a matrix
Given this $n \times n$ matrix:
$$ A= \left(\begin{matrix}a_1&a_1&...&a_1\\a_1&a_2&...&a_2\\\vdots& &\ddots &\vdots\\a_1&a_2&...&a_n\end{matrix}\right) $...
0
votes
1answer
59 views
LU composition of a tridiagonal matrix
Given a triadiagonal Matrix A with
$$A = \begin{bmatrix}
d_1 &e1\\
c_2 & d_2 & e_2 \\
& c_3 & \ddots & \ddots \\
& & \ddots & \ddots & e_{n-1} \\
& & &...
0
votes
1answer
41 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.
$$
\left[
\begin{matrix}
t_0*d_0 & -t_1*e_0 & 0 & 0 & 0 \\
0 & t_1*d_1 & -t_2*e_1 & 0 &...
1
vote
0answers
50 views
Schur complement in LUP decomposition of a block tridiagonal matrix
Section 2.2 of the article On twisted factorizations of block tridiagonal matrices explains how to do a LUP decomposition of block tridiagonal matrices by showing the process on a 4 blocks by 4 blocks ...
15
votes
3answers
2k views
Are these square matrices always diagonalisable?
When trying to solve a physics problem on decoupling a system of ODEs, I found myself needing to address the following problem:
Let $A_n\in M_n(\mathbb R)$ be the matrix with all $1$s above its ...
7
votes
1answer
235 views
Finding the determinant of a tridiagonal matrix
$$\begin{vmatrix}x&1&0&0&āÆ\\-n&x-2&2&0&āÆ\\0&-(n-1)&x-4&3&āÆ\\ā®&ā±&ā±&ā±&ā®\\0&āÆ&-2&x-2(n-1)&n\\0&0&āÆ&-1&x-2n\...
1
vote
1answer
216 views
Inverse of symmetric tridiagonal block Toeplitz matrix
There is a triagonal block matrix $M$ of form:
$$
M = \begin{bmatrix}
A & B^T & 0 & 0 & \cdots & 0 & 0 \\
B & A & B^T & 0 & \cdots & 0 & 0 \\
0 & B ...
1
vote
1answer
192 views
Determinants of tridiagonal matrices
A square matrix $A = [a_{ij}]$ is called ${\bf tridiagonal}$ if
$a_{ij}=0$ for $|i-j|>1$. Try to guess a formula for the determinant
of tridiagonal matrix, say $a_i = a_{ii}$ for $i=1,...,n$, $b_i =...
3
votes
3answers
614 views
Inverse of tridiagonal Toeplitz matrix
Consider the following tridiagonal Toeplitz matrix. Let $n$ be even.
$${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}}
{0}&{1}&{}&{}&{}\\
{1}&{0}&{1}&{}&{}\\
{}&...
0
votes
3answers
103 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
659 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
391 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
114 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
95 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 ...
