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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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 ...
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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 ...
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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{...
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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 \\ & & & ...
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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 &...
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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 ...
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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 ...
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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 ...
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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 &...
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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 &...
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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 ...
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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 &...
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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: $$ \pd{^2u}{t^2}(x,t)=c^2\pd{^2u}{x^2}(x,t),~~ t>0,~x\in(0,1)\tag4 $$ ...
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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 \\...
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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, \...
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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 &...
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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 ...
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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 ...
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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} ...
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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}^{(...
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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 \...
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0answers
38 views

Lanczos on structured matrix

Suppose $T ∈ \mathbb R^{n\times n}$ is tridiagonal and symmetric and that $v∈\mathbb R^n$. Show how the Lanczos process can be used (in principle) to compute an orthogonal $Q ∈ \mathbb R^{n\times n}$ ...
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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 & \...
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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 ...
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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 ...
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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} & {\...
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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 ...
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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
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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) ...
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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.
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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&-...
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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 = ...
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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(...
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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
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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 ...
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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
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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) $...
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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} \\ & & &...
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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 &...
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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 ...
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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
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1answer
235 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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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
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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
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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}&{}&{}\\ {}&...
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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{...
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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 ...
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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 ...
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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
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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 ...