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
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 ...
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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{\...
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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 ...
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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 ...
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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 & ...
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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 ...
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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 ...
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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 (...
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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 , \...
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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 ...
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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 ...
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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. ...
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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 $L$ ...
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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\\ ...
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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 ...
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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 ...
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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 ...
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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_{...
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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}...
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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 &...
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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&...
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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 $0$ ...
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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\\&...
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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 $\det(A),\det(B),n$ ...
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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 ...
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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 ‎&...
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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 ...
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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 ...
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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& ‎\...
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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'...
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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 & ...
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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 ...
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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.
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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 &...
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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 & ...
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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 ...
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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 ...
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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{...
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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 \\ & & & ...
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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 &...
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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 ...
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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 ...
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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 &...
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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 ...
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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: $$ \pd{^2u}{t^2}(x,t)=c^2\pd{^2u}{x^2}(x,t),~~ t>0,~x\in(0,1)\tag4 $$ ...
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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 \\...
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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, \...
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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 &...
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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 ...
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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 ...