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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Give algorithm for finding bidiagonal matrix similar to triangular matrix

Is there an algorithm that takes an upper triangular matrix $T$ over the complex numbers as input, and outputs a bidiagonal matrix $B$ which $T$ is similar to? We may assume that $T$ is invertible. ...
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Leading eigenvalue of a strange tridiagonal matrix with matrix sub-blocks

Let's assume the following crazy matrix \begin{equation} P = \begin{pmatrix} \mathbb{0}_{1\times 1} & \alpha \mathbb{1}_{1\times N} & \mathbb{0}_{1\times\frac{N(N-7)}{2!}} & \...
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Linear phase shifts in the exponential of special tridiagonal matrices

I've been working on a physics problem that has led me to make the following numerical observation. Let $L$ be an $N\times N$ real-symmetric tridiagonal matrix whose diagonal entries are zero. This ...
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Is the product $B D B^T$ always a symmetric tridiagonal matrix? Where $D$ is a diagonal matrix and $B$ a sparse matrix.

I have a diagonal matrix ${\bf D}_{n \times n}$ and a rectangular matrix ${\bf B}_{m \times n}$ where $n \gg m$. All but $m$ rows of ${\bf B}$ have non-zero elements. These $m$ rows have only six non-...
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Eigenvalues of tridiagonal Toeplitz matrix with diagonals $1$, $0$, and $-1$

Consider a matrix $A \in M_n(\mathbb{R})$ with entries denoted by $A=[a_{ij}]$. When $i=j+1$, $a_{ij}=1$, and when $i=j-1$, $a_{ij}=-1$, with all other entries being zero. Determine the eigenvalues of ...
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What are the eigenvalues of a tridiagonal Toeplitz matrix?

Consider a matrix $M \in \mathbb{R}^{n \times n}$ in the form: $$ M = \begin{bmatrix} \alpha & \beta & 0 & \cdots & 0 \\ \gamma & \alpha & \...
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Show that the elements of the LR decomposition for tridiagonal matrices T can be determined by the following recursive relation

Show that the elements of the LR decomposition $ T=L R $ can be expressed by tridiagonal matrices $ T $ with can be determined by the following recursive relation: How could you prove this statement ...
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Diagonalization matrix and relate to abstract algebra

A matrix $A\in M_n(K)$ is diagonalizable iff $\exists\hspace{.03cm}P\in GL_n(K):\ P^{-1}AP$ is diagonal matrix. Then $B=P^{-1}AP$ is called similar to $A$. I realize this is the conjugate element ...
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If $f:\ V\longrightarrow V$ be linear then how the fact $V=ker\,f+im\,f$ relate to the diagonalization / Jordanization of $f$?

Let $V$ be a n-dimensions vector space and $f$ be a linear map on $V$. Then is there any relevant between $V=ker\,f+im\,f$ and the diagonalization/Jordanization of $f$ ? Thanks
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Unsymmetric to Symmetric Tridiagonal Matrix

According to the Wikipedia page here, a real, unsymmetric tridiagonal matrix can be brought to symmetric form by a similarity transform. Does anyone know if a generalization of the formula given there ...
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Intriguing Tridiagonal Matrix

Given a positive sequence $\{a_n\}_{n=1}^N$, please consider the matrix $$ \begin{bmatrix} a_1 & -a_1 & & & & \\ -a_1 & a_2 + a_1 & -a_2 & & & \\ & - ...
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Eigenvalues of a symmetric tridiagonal matrix

I'm looking for the eigenvalues of the following symmetric tridiagonal matrix \begin{pmatrix} a & z & 0 & 0 & 0 \\ z & b & z & 0 & 0 \\ 0 & z & 0 & z & ...
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Calculating the characteristic polynomial of a block tridiagonal Toeplitz Symmetric matrix

I am trying to calculate the characteristic polynomial of a block tridiagonal matrix and I need some help. This matrix is a representation of a tight-binding Hamiltonian of a finite grid of graphene, ...
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Diagonalization of specific symmetric tridiagonal matrix

I am wondering if there is a way to get an explicit expression for the eigenvalues (and possibly the eigenvectors) of a symmetric tridiagonal matrix with the following peculiar structure: $$ \left( \...
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Closed form of the eigenvalues of a specific tridiagonal matrix

Hello I would like to know if someone has an idea of how to compute the eigenvalues (in a closed form) of the following matrix in order to implement it for fast calculations. The matrix is the ...
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Inverse of block tridiagonal positive definite symmetric matrix

I have to compute the following matrix inverse $$ \left( \begin{bmatrix}2\bf{B} & -\bf{B} & \bf{0} \\ -\bf{B} & 2\bf{B}& -\bf{B} \\\bf{0} & -\bf{B}& \bf{B} \end{bmatrix} + \...
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Eigenvalue bound for a entrywise bounded trigonal positive definite matrix

Given I have a matrix $A\in\mathbb{R}^{n\times n}$ and it is tridiagonal and positive definite such that $b_{ij}$ must be zero if $|i-j|>1$. Furthermore, $0<C_1<a_{ij}<C_2$ for $|i-j|\le 1$...
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If $A$ is a symmetric tridiagonal matrix, then $\det(A)=a_{11}\det\left(M_{11}\right)-a_{12}^2\det(B)$

From Section 2.1 in Steven J. Leon & Lisette de Pillis's Linear Algebra with Applications (10th edition): Problem 13. Let $A$ be a symmetric tridiagonal matrix (i.e. $A$ is symmetric and $a_{ij} =...
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Localization of eigenvalues for block-tridiagonal Hermitian Toeplitz matrix made of gamma blocks

I am studying the spectrum of a particular kind of block-tridiagonal Hermitian Toeplitz matrix made of three bands $\{B,A,C\}$ $$ T_n = \begin{pmatrix} A & C & 0 & \dots & 0\\ ...
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How can I numerically obtain the eigenvalues of the second-order derivative from the discretization matrix?

I want to find the eigenvalues of the second-order derivative (with pure Neumann boundary conditions, on the interval $[0,1]$) numerically and compare them with the eigenvalues derived analytically. ...
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Does there exist a unitary matrix such that $Uv=\|v\|e_{1}$ for $v\in\Bbb{C}^{n}$?

Let $v\in\Bbb{C}^{n}$ be a fixed vector. Then does there exist a unitary Matrix $U$ such that $Uv=\|v\|e_{1}$? or even $Uv=c\|v\|e_{1}$ for some real constant $c$? I am looking for a Householder ...
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Proving that a symmetric tridiagonal matrix is positive definite

How to prove that the square and symmetric matrix below is positive definite? $$A = \begin{bmatrix} 81 & -40 & \\ -40 & 101 & -40 & \\ 0 & -40 & 101 & -...
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Eigenvalues of an $n \times n$ symmetric tridiagonal matrix

How to show that an $n \times n$ tridiagonal matrix only has positive eigenvalues? $$ \begin{pmatrix} 70 & -35 & 0 & \dots & 0 \\\ -35 & 120 & -35 & \ddots & \...
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Computational advantage for tensor product operators

Consider a real-symmetric square matrix, $H$, which we take to be "large" for computational purposes - say, $10^{4}$ rows and $10^{4}$ columns. Such a matrix can be diagonalized using ...
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Determinant of a tridiagonal matrix with constant values $5,3,2$ on the main, upper and lower diagonals [duplicate]

Can you help me to compute determinant of matrix $$A =\begin{pmatrix} 5 &3 &0 &\cdots &0 &0 \\ 2&5 &3 &\cdots &0 &0 \\ 0&2 &5 &\...
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How does accuracy of eigenvalues from lanczos algorithm change with tridiagonal matrix size

Trying to implement Lanczos algorithm, to find the lowest $M$ eigenvalues of a very large $N$ by $N$ matrix $H$. I have implemented a working algorithm, which lets me find matrix $V$ with orthonormal ...
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No two successive leading principal minors of tridiagonal matrices share common eigenvalues

I would like to know if this is a correct proof to show why no two successive leading principal minors of a tridiagonal matrix have common eigenvalues. Let's assume that there exist two successive ...
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sum of the elements of a tridiagonal matrix and its inverse

I have a tridiagonal matrix $A$ of dimension $n$ with terms $1+\theta^2$ on the diagonal and $\theta$ on the secondary diagonals: $A=\begin{bmatrix}1+\theta^2 & \theta & 0 &...& 0\\ \...
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Eigenvalues of symmetric tridiagonal matrices with complex entries

In this paper the authors proved that for a real symmetric tridiagonal matrix $T_n$, where $b_i \neq 0$, as follows $$T_n = \begin{bmatrix} a_1&b_1&0&0&0&0&0&0&\cdots&...
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Positive definite matrices - symmetric - diagonal elements

a) Show that the tridiagonal matrix $$A=\begin{pmatrix} 2 & 1 & \dots & 0 \\ 1 & 2 & 1 & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \ldots & 1 & ...
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Are tridiagonal stochastic matrices irreducible?

According to Wikipedia, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal and the supradiagonal. To my understanding, in a tridiagonal ...
Jack's user avatar
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The smallest positive eigenvalue of a special triangle matrix

Suppose I have a $n\times n$ symmetric tridiagonal matrix: $$ A = \left(\begin{array}{ccccc} 1 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & & \cdots \\ 0 & -1 & 2 &...
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Conditions to bound the eigenvalues of a symmetric tridiagonal matrix

Consider an arbitrary $n\times n$ symmetric tridiagonal matrix $M_n$ with elements $$(M_n)_{ij} = \begin{cases} 0, \quad i=j \\ u_{\text{min(i,j)}}, \quad |i-j|=1 \\ 0, \quad \text{else} \end{cases}$$ ...
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Matrix-vector norm inequalities in tridiagonal matrix

Let's consider the linear system $A\vec{x}=\vec{b}$, with $A$ being a diagonally dominant by columns and tridiagonal matrix, that is: $$A=\begin{pmatrix} d_1&& c_2 && &&...
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Tridiagonal matrix power with negative diagonal

I have a tridiagonal matrix with negative diagonal and and non-negative off-diagonal elements where every row sums to zero (it is the generator to an M/M/n/n queue). My experiments indicate that $|A|^...
Andreas Dahlberg's user avatar
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Maximal and minimal eigenvalues of a symmetric tridiagonal Toeplitz matrix

Given $m \times m$ symmetric tridiagonal Toeplitz matrices $$M=\begin{pmatrix} 4 & 1 & & \\ 1 & 4 & \ddots & \\ & \ddots & \ddots & 1\\ & & 1 & 4\...
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Proving a complex block tridiagonal matrix is nonsingular

I have a complex block tridiagonal matrix that I am trying to prove has an inverse. The matrix M here: $$M = \begin{bmatrix} A' & B' & 0 & \cdots & 0 \\ B & A & B & \...
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Inversion of a Toeplitz tridiagonal matrix

Currently, I am working on solving a PDE using the finite element method (FEM) and facing the problem of finding the inverse of the following Toeplitz tridiagonal matrix $$\mathbf{M} = \begin{pmatrix} ...
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Tridiagonal matrix with main diagonal equal to 1, inverse 1-norm upper bound

Description Suppose we have the following non singular tridiagonal matrix $$ B = \begin{bmatrix} 1 & a_1 \\ b_2 & 1 & a_2 \\ & \ddots & \ddots & \ddots \\ && b_{n-...
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Recursive computation of determinant of Toeplitz tridiagonal matrix

Let a matrix be a tridiagonal matrix of size $n \times n$, with elements equal to $2$ on the main diagonal, elements equal to $1$ directly above the main diagonal, elements equal to $3$ directly below ...
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Using matrix to analyze an ODE system $\begin{cases}\dot a_k(t)=2(b_k^2-b_{k-1}^2)\\\dot b_k(t)=b_k(a_{k+1}-a_{k})\end{cases}$ with $b_0(t)=b_n(t)=0$.

Let $a_k(t), b_k (t) \in \mathbb{R} \ (k = 1, 2, \dots, n)$ satisfy the following differential equations: $$\begin{aligned} \frac{d}{dt}a_k(t) &= 2 \left( b_k^2 - b_{k-1}^2 \right) \\ \frac{d}{dt} ...
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Toeplitz tridiagonal matrix with $0$s on main diagonal and $1$s on sub/superdiagonal has distinct eigenvalues [closed]

$$\begin{pmatrix}0&1&&&\\ 1&\ddots&\ddots&&\\ &\ddots&\ddots&\ddots&\\ &&\ddots&\ddots&1\\ &&&1&0\end{pmatrix}$$ has ...
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Can this $3 \times 3$ tridiagonal Toeplitz matrix be rank-$1$?

I am trying to determine whether the following tridiagonal $3 \times 3$ matrix can have a rank of $1$. $$\begin{bmatrix}a&b&0\\b&a&b\\0&b&a\end{bmatrix}$$ For $a = b = 0$, the ...
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Proving that every symmetric matrix is similar to a symmetric tridiagonal matrix

If we consider $A$ to be a symmetric $n\times n$ matrix, then how can we show that there exists a unitary matrix $G$ such that : $$ A=GTG^{*} $$ where $T$ is a tridiagonal symmetric matrix. I am not ...
Альбина's user avatar
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How to notate off-diagonal blocks for a tridiagonal block matrix

I have a block diagonal matrix that also has partitioned identity matrices in the off-diagonal blocks, like a tridiagonal block matrix. It looks something like $$ \begin{bmatrix} \textbf{A} & I &...
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Doubt in a concept while finding the determinant of a tridiagonal matrix?

I was going through the second answer of this question but I can't understand if it is correct to expand the determinant along the column 1st and then in the recursive step to compute the determinant ...
ThirstForMaths's user avatar
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Analytical expression for the determinant of block tridiagonal matrix

I have a $3n\times3n$ matrix $M$ that is in the following block tridiagonal form: $$M=\begin{pmatrix} A & B^T & 0\\ B & A & UBU \\ 0 & UB^T U & A\\ \end{pmatrix}$$ where $A,B,...
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How to find the eigenvalues of a block tridiagonal Toeplitz matrix?

I have a block tridiagonal Toeplitz matrix $$M=\begin{bmatrix} A & Z & O\\ Y & A & Z\\ O & Y & A\end{bmatrix}$$ where $$A=\begin{bmatrix} 0 & 1 & 0 & 1\\ 1 & 0 &...
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Eigenvalues of a certain symmetric tridiagonal Toeplitz matrix

Is there any way that can explicitly calculate eigenvalues (or at least the largest eigenvalue) of the following $n \times n$ symmetric matrix: \begin{pmatrix} 1 & 1 & 0 & 0 & \cdots \\...
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Parallel LU decomposition of diagonal matrices

I have seen efficient implementations of LU decomposition for tridiagonal and penta diagonal matrices. I have a banded diagonal matrix that is huge 100k rows by 100k columns, but each row has only 27 ...
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