Questions tagged [tridiagonal-matrices]
Relating to all $n\times n$ matrices $(A)$ with the property $a_{i,j}=0$ if $|j-i|>1$
158
questions
0
votes
1answer
31 views
Instability of cubic splines
I'm supposed to show, that the calculation of cubic splines with tridiagonal matrices is unstable. To show this I'm supposed to consider the function $s_1: [x_0, x_n] \to \mathbb{R}$ which is a spline ...
0
votes
1answer
38 views
Eigenvalue Problem of a hermitian tridiagonal matrix
I want to prove that the eigenvalues of the following complex, hermitian matrices are the same
$$
A=\begin{pmatrix}
\alpha_1&\beta_1\\
\overline{\beta_1}&\alpha_2&\beta_2\\
&\overline{\...
0
votes
0answers
17 views
Cholesky factoriation of repeatitive block tridiagonal matrix
I have a square symmetric positive definite 2n x 2n matrix $Q$. $Q$ is partitioned into 4 square sub-matrices of equal size (n x n)
$$Q = \begin{bmatrix} A & B^T \\ B & C \end{bmatrix}$$
I ...
0
votes
2answers
67 views
Find eigenvalues and eigenvectors of particular Toeplitz matrix
Assume a matrix in this form:
$$
\begin{bmatrix}
b & c & 0 & \dots & 0 & a \\
a & b & c & 0 ...
0
votes
0answers
43 views
Eigendecomposition of symmetric block tridiagonal matrix with symmetric tridiagonal blocks
I have a symmetric block tridiagonal matrix of the form:
$$ K = \begin{bmatrix} C_1 & D_1 & 0 & 0 & 0 & \cdots \\ D_1 & C_2 & C_2 & 0 & 0 & \cdots \\ 0 & ...
0
votes
0answers
19 views
Advection-Diffusion Equation with two variables
Consider the following equation:
$\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial x^2} -u \frac{\partial T}{\partial x} -T \frac{\partial u}{\partial x}$
where $T$ is the ...
0
votes
1answer
40 views
What is the name of the theorem of tridiagonal reduction of symmetric matrices?
I have over the last year been implementing a code to solve the many-body Schrödinger equation for applications in Nuclear physics, as part of my PhD-studies. The most important step in that code is ...
0
votes
0answers
24 views
generalized eigenvalue problem with tri-diagonal matrix admits only real eigenvalues
I am considering this generalized eigenvalue problem: $( A-\lambda B ) x = \mathbf{0}$, here $A$ and $B$ are real tri-diagonal matrix and $A$ is symmetric but $B$ is not . I find this problem usually (...
1
vote
0answers
58 views
Proof involving tridiagonal diagonally dominant matrix
$\mathbf{Background}$
Define a finite difference grid on the rectangle $[0,X] \times [0,T]$,
$(x_j,t_n) = (x_0 + j\Delta x,n\Delta t) \text { for } 0 \leq j \leq M \text { and } 0 \leq n \leq N , \...
1
vote
0answers
16 views
Determining the eigenvalues of the intersection matrix of a Johnson association scheme
The intersection matrix of the Johnson association scheme is discussed in Ito-Bannai, Algebraic Combinatorics I, page 211, (2.16).
I am not an expert in the area. The question is: can you point me to ...
5
votes
1answer
51 views
since a symmetric tridiagonal matrix contains only two distinct vectors
I don't understand meaning of "since a symmetric tridiagonal matrix contains only two distinct vectors" I write example for symmetric tridiagonal matrix. The 1st row vector and 1st column ...
0
votes
1answer
26 views
How can one compute the determinant of a tridiagonal matrix when using integers?
My question is similar to How to compute the determinant of a tridiagonal matrix with constant diagonals?
However, all of the options seem to include computations that are not permitted for integers. ...
1
vote
0answers
36 views
Example of regular tridiagonal matrix $A$ with given properties
I am looking for a regular tridiagonal matrix $A$ such that at the LU-decomposition with partial column pivoting the matrices $L$ and $U$ are also tridiagonal, but with total pivoting the matrices $L$ ...
-1
votes
1answer
122 views
Supremum of $\| A \|$ [closed]
I have the following tridiagonal matrix :
$$A = \begin{pmatrix}
9 & 1 & 0 &\dots & 0\\
1 & 9 & 1 & \ddots & \vdots \\
0 & 1 & \ddots & \ddots & 0\\
...
0
votes
0answers
43 views
linear transformation, is diagonalizable? all values of a?
People i have the next problem.
Let $f: \mathbb R_3 → \mathbb R_3$ be the linear transformation such that https://i.stack.imgur.com/qLTqd.png with $\mathcal B = \{(1,0, −1), (0,0,1), (0,1 , 0)\}$. The ...
1
vote
0answers
33 views
Finding the eigenvalues of a discrete laplacian on an infinite lattice
If we define the Laplacian as a square matrix with zeroes on the diagonal, and -1 on the diagonals exactly above and below the main diagonal, and 0 everywhere else, how would one go about finding its ...
0
votes
0answers
45 views
Show that $Q$ is Hessenberg for QR Factorization of tridiagonal matrices
Let $T$ be a tridiagonal, symmetric matrix in $\mathbb{R}^{n\times n}$. The QR algorithm of $T$ with shifts is defined is as follows;
$$T^{(k)} − \mu I = Q^{(k)}R^{(k)}$$
(where right-hand side is a ...
0
votes
0answers
33 views
How to show that two Hermitian tridiagonal matrices are similar?
Given two tridiagonal hermitian matrices A,B with $a_i\in \mathbb{R}$ and $b_i\in \mathbb{C}$ as follows
\begin{align}
A=
\begin{pmatrix}
a_{1} & |b_1| & \cdots & 0 \\
|b_1| & a_{...
3
votes
0answers
57 views
Decomposition of a dense matrix into $m$ tridiagonal matrices
Given an $n\times n$ dense matrix $\mathbf{K}$, is there any decomposition that factorizes $\mathbf{K}$ into $m$ tridiagonal matrices such that $\mathbf{K} = \mathbf{B}_1 \, \mathbf{B}_2 \, \mathbf{B}...
0
votes
1answer
53 views
Diagonalize a quasi-uniform tridiagonal matrix
How to diagonalize the following matrix?
$$
A=\begin{pmatrix}
0 & a& 0 & 0 & 0 & \cdots \\
a& 0 & b & 0 & 0 & \cdots \\
0 & b & 0 & b & 0 &...
0
votes
0answers
52 views
Eigenvalues of sum of a tridiagonal matrix and another matrix whose square is a diagonal matrix
$A$ is a tridiagonal matrix of the form $$\left( \begin{array}{ccccccccc}d_1&a&0&0&\cdots&0&0&0&0\\a&d_2&a&0&\cdots&0&0&0&0\\0&a&...
0
votes
1answer
57 views
Show that the eigenvalues of the matrix are symmetric around origin
Let $A \in M_n (\mathbb{R})$ be a matrix whose all entries are $0$ except those on the diagonals above and below the principal diagonal, which are all $1$. That is, $a_{ij} = 1$ if $|i-j|=1$ and $0$ ...
0
votes
0answers
45 views
How to find the eigenvalues of the following tridiagonal matrix [duplicate]
The $n$-order tridiagonal matrix is defined by $A(i,i)=0, A(i,i+1)=n-i, A(i+1,i)=i$, i.e.,
$$A=\begin{pmatrix}0&n-1\\1&0&n-2\\&2&0&n-3\\&&\ddots&\ddots&\ddots\\&...
1
vote
0answers
72 views
Determinant of the sum of tridiagonal and anti-tridiagonal matrices
$A$ is a tridiagonal matrix and $B$ is an anti-tridiagonal matrix, both of size $n\times n$, such that $B^2$ is a diagonal matrix. Is it possible to express $\det(A+B)$ in terms of $\det(A),\det(B),n$ ...
0
votes
0answers
21 views
computing the power of a tridiagonal matrix in R: exact solution and implementation
I am trying to solve a tridiagonal matrix power in R, as shown below. This a tridiagonal matrix at the power of 't'. It can be interpreted as a classic transition matrix in a Markov chain.
The paper ...
2
votes
0answers
44 views
similarty anti-tridiagonal?
Let $n \times n$ matrix $H_n$ be anti-tridiagonal. For example, when $n = 4$,
$$H_4 = \begin{bmatrix} 0 & 0 & a & b\\ 0 & a & b & c\\ a & b & c & 0\\ b &...
0
votes
0answers
100 views
Inverting a block tridiagonal matrix
It's my first question over here so if there is anything that is unclear please let me know.
My question concerns the inversion of a block tridiagonal symmetric matrix. I have checked multiple papers ...
3
votes
0answers
65 views
Falkner Skan solution using a tridiagonal matrix method. For 0, -ve $\beta$ I get incorrect solutions?
I am trying to solve the Falkner Skan equation for wedge flows numerically, by first reducing it to a second order system and then solving a tridiagonal matrix equation iteratively till solution ...
0
votes
1answer
57 views
similarity matrix tridiagonal
I am solving a problem, in the middle of which I reached the following block-tridiagonal matrix.
\begin{bmatrix}
0 & B_1 & &0 \\
B_1 & 0 & \ddots \\
&\ddots& \...
1
vote
0answers
71 views
Eigenvalues of a tridiagonal matrix with $-1, 2, -1$ as entries
This problem was given as an exercise for section 6.2 of Hoffman & Kunze's book in a course in linear algebra, which I couldn't solve and I really don't know how to solve it, may be because I don'...
6
votes
2answers
238 views
Calculate determinant of a continuant matrix with variable elements?
One of the problems in Muir's Treatise on determinants in exercise set XXX asks to evaluate the determinant of a continuant matrix
$$
f_n(x,y)=\left|\begin{array}{cccc}
x & 1 & 0 & 0 & ...
0
votes
0answers
33 views
Tridiagonal matrices and vector space dimension
"The set of all $n$ × $n$ tridiagonal matrices forms a 3n-2 dimensional vector space."(Wikipedia)
I'm familiar with what a tridiagonal matrix is but I'm not sure why this specifically is ...
0
votes
1answer
51 views
What is a simple definition of Lanczos iteration that is understandable?
I have seen that Lanczos's Algorithm can be used to tri-diagonalize a matrix but all of the definitions I have seen of it have been very complicated to understand.
1
vote
1answer
44 views
Special tridiagonal matrix name and property
I was wondering if there is a special name for the following kind of tridiagonal matrices ? And if yes, are there any books or articles which talk about their properties ?
\begin{pmatrix}
\alpha_1 &...
3
votes
1answer
72 views
Elementary block matrix operations to block-triangularize block tridiagonal matrix
For some flow modeling purpose, the system $(S)$ defined as $MX=L$ is solved, where $M$ is a tridiagonal block matrix defined as
$$ M= \begin{pmatrix}B_1& C_1 &0 &0 \\ A_2 & B_2 & ...
2
votes
1answer
180 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
34 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
28 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
40 views
Signature of a tridiagonal matrix
Let $A$ be a tridiagonal matrix of the form
$$A=
\begin{pmatrix}
a_1 & 1 \\
1 & a_2 & 1 \\
& 1 & \ddots & \ddots \\
& & \ddots & \ddots & 1 \\
& & & ...
0
votes
1answer
58 views
Why does tridiagonal matrix reduce noise?
Let
$$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
24 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
19 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 ...
1
vote
0answers
63 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,
$$
\begin{pmatrix}
R_{\phantom{1}} & R_{1} & 0 & 0 & 0 &...
1
vote
0answers
38 views
Why does Relative Robust Representations (RRR) of Matrices determine eigenvalues to high relative accuracy?
I am trying to understand the MRRR algorithm for finding eigenpairs. As part of this I am reading Parletts and Dhillons paper on "Relatively robust representations of symmetric tridiagonals". I am ...
0
votes
1answer
42 views
Numerical methods, show that wave equation expression is constant
$\newcommand{\pd}[2]{\frac{\partial#1}{\partial#2}}
\newcommand{\IR}{\Bbb R}\newcommand{\dd}{\mathrm{d}}$
The wave equation:
$$
\pd{^2u}{t^2}(x,t)=c^2\pd{^2u}{x^2}(x,t),~~ t>0,~x\in(0,1)\tag4
$$
...
1
vote
1answer
54 views
Show that $\det A_n = \sum_{i=0}^{n}a^{2i}$
Let $\mathbb{F}$ be a field and $a\in\mathbb{F}$. For $n \geq 1$ let
$$A_n=\begin{pmatrix} 1+a^2 & a & 0 & 0 & \ldots & 0 \\ a & 1+a^2 & a & 0 & \ldots & 0 \\...
1
vote
0answers
32 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
85 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
285 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
59 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
...
