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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How to show two matrices are similar? [closed]

Show that the matrices A = ⎡ 1 1 1 ⎤ ⎢ 1 1 1 ⎥ ⎣ 1 1 1 ⎦ and B = ⎡ 3 0 0 ⎤ ⎢ 0 0 0 ⎥ ⎣ 0 0 0 ⎦ are similar? I know I should show there exist P- such that A=PBP-, but how do I find P, and is there a ...
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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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Trace of the square root of a tridiagonal matrix

I have the following $n \times n$ symmetric tridiagonal matrix: \begin{equation} \label{myeq} M = \left[ \begin{matrix} a_{1}^{2} & -ca_{1} & & & & 0 \\ -ca_{1} & a_{2}^{2} &...
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inverting a tridiagonal circulant matrix with alternating elements

First of all, I want to underline that my knowledge regarding matrices is extremely restricted.. It just so happens that they are popping out everywhere in a recent project of mine. So, in this ...
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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 the property of an ODE?

Let $a_k(t), b_k (t) \in \mathbb{R} \ (k = 1, 2, \dots, n)$ satisfy the following differential equations: $$\begin{aligned} \dot a_k(t) &= 2 \left( b_k^2 - b_{k-1}^2 \right) \\ \dot b_k(t) &= ...
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Decomposition of stochastic matrices into tridiagonal ones

I was wondering if some results were known for the following problem for which I couldn't find any information, except for this similar question that unfortunately doesn't have answers. So let's start ...
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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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Wilkinson's matrix eigengap

Task The $(2n+1) \times (2n+1)$ Wilkinson matrix is a symmetric tridiagonal matrix whose main diagonal is $$n, n-1, \ldots, 1, 0, 1, \ldots, n$$ and whose entries on the first diagonals above and ...
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the spectrum of tridiagonal matrix is bounded

What is the smallest eigenvalue of this infinite, symmetric, tridiagonal matrix? how could I prove that tridiagonal matrix have the smallest eigenvalue,and for any 𝜆 , the spectrum of that matrix is ...
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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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Positive definiteness of symmetric block tridiagonal matrix

Let $A\in\mathbb{R}^{n\times n}$ be the symmetric tridiagonal matrix \begin{equation*} A:=\begin{pmatrix} u_1 & v_1 & \\ v_1 & u_2 & v_2 \\ & v_2 & \ddots & \ddots \\ &...
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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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What is the smallest eigenvalue of this infinite, symmetric, tridiagonal matrix?

$A_{nm}$ ($n,m = 0, 1, 2, \ldots$) is a symmetric, tridiagonal matrix. The diagonal elements are $A_{nn} = a_n = n + 1$, and the off-diagonal elements are $A_{n,n+1} = A_{n+1,n} = b_n = \lambda \sqrt{\...
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Partial derivative of a diagonal matrix w.r.t a vector

I am trying to find the second partial derivative of the function $Y=diag\boldsymbol(S)\mathbb P diag \boldsymbol(\beta)diag^{-1}(\mathbb P^{T}\boldsymbol S +\mathbb P^{T}\boldsymbol E + \mathbb P^{T}\...
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Exponential of infinite dimensional matrix

Asked this on MathOverflow with some more details I have a matrix originating from Master Equation for birth death process on semi infinite lattice. It is tridiagonal. It is not symmetric. I wanted ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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3 votes
1 answer
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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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