Questions tagged [tridiagonal-matrices]
A tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.
35 questions
2
votes
0
answers
103
views
Polynomial filter on symmetric tridiagonal matrices
During my research, I encountered a problem involving symmetric tridiagonal matrices and their eigenvectors, which I have been attempting to solve since then. So far, I have only managed to obtain a ...
- 61
1
vote
1
answer
277
views
Exact form of eigenvalues of pentadiagonal Toeplitz matrices
The tridiagonal Toeplitz matrices
$$\begin{pmatrix}
a & b & & \\
c & \ddots & \ddots \\
& \ddots & \ddots & b \\
& & c ...
2
votes
0
answers
114
views
Is it possible to analyze the eigenvalue of a specific tridiagonal matrix?
I'm considering the $n \times n$ tridiagonal matrix
$$ A = \begin{pmatrix}
0 & 1 & & & \\
1 & c & 1 & & \\
...
- 21
0
votes
1
answer
139
views
Eigenvectors for specific eigenvalues
When reading the following paper:
Ed S. Coakley, Vladimir Rokhlin, A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices, Applied and Computational ...
2
votes
1
answer
262
views
Diagonalize almost symmetric tridiagonal matrix
I begin with an $n \times n$ real symmetric tridiagonal matrix. However, I replace the non-zero elements in the first and last rows with zeros, so it is no longer symmetric
$$M = \begin{bmatrix} 0 &...
- 141
3
votes
2
answers
483
views
Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix
The question is the following: given a matrix
$$A=\begin{pmatrix}
1& 2 & & & & \\
1& 0& 1 & & & \\
& 1& 0& 1 & &\\
& &...
- 145
1
vote
0
answers
175
views
Solving a block tridiagonal system with diagonal perturbations
Say we have a block tridiagonal matrix, $T \in \mathbb{R}^{NL \times NL}$, with constant off diagonals, $\mathbf{B} \in \mathbb{R}^{L\times L}$, given by
$$
T = \begin{bmatrix} \mathbf{A}_1 & \...
- 11
3
votes
0
answers
709
views
Diagonalizing a block tridiagonal matrix
Is there an efficient way to diagonalize a block tridiagonal $N\times N$ matrix of the following form:
\begin{matrix}
A_0 & B & 0 & 0 & \ldots \\
B & A_1 & B & 0 & \...
- 31
0
votes
0
answers
209
views
Highest eigenvalue of symmetric tridiagonal matrix
I was numerically playing with tridiagonal symmetric matrix (zero on diagonal) of the form
\begin{pmatrix}
0 & b_1 & 0 & 0 & 0 & \ldots & 0 \\
b_1 & 0 & b_2 & 0 &...
- 101
4
votes
0
answers
388
views
Exponential of infinite dimensional matrix
Originally posted on Math SE but didn't get any responses. Thus, I thought I would ask here with some more details.
I have a matrix originating from Master Equation for birth death process on semi ...
- 41
3
votes
1
answer
502
views
Derivative of eigenvalues of a symmetric tridiagonal matrix built via the Lanczos-Arnoldi scheme
Suppose $\mathbf{A}(\mu)$ being a symmetric positive definite matrix of dimension $n$ where its elements depend parametrically on the real parameter $\mu$.
Suppose now to build the orthonormal basis ...
- 131
5
votes
2
answers
1k
views
Sufficient conditions for invertibility of a block tridiagonal matrix
Let $M_n \in \mathbb{R}^{N \times N}$ be a block-tridiagonal matrix:
$$M_n = \begin{bmatrix} B_1 & C_1 & 0 & 0 & \cdots & 0 \\ A_1 & B_2 & C_2 & 0 & \cdots & 0 \...
- 397
5
votes
0
answers
180
views
Is there a discrete Schrödinger operator with empty spectrum?
A relatively well-known example of (continuous) Schrödinger operator with empty spectrum is the complex Airy operator on the line, i.e., the operator acting on $L^{2}(\mathbb{R})$ given by the ...
- 2,208
1
vote
0
answers
544
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)
$$
\begin{...
- 111
3
votes
1
answer
2k
views
Eigenvalue distribution of a block tridiagonal matrix
Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i-1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$.
For some positive integer $k$, I define ...
- 924
2
votes
2
answers
348
views
Is tridiagonal reduction the current best practice to compute eigenvalues of random matrices from the Gaussian ensembles (GOE, GUE, GSE)?
I have tried to compute the eigenvalues of random matrices of the GOE ensemble, using MATLAB.
Such matrices of size $n * n $ can be obtained easily, symmetrizing matrices whose elements follow the ...
- 153
2
votes
1
answer
656
views
Using permutation matrix to convert a matrix into tridiagonal matrix [closed]
Let $A \in \mathbb{R}^{n \times n}$ be a bidiagonal matrix with non zero elements on its diagonal and super diagonal. Let $B$ be defined as
$$B=\begin{bmatrix}0&{A} \\{A}^T &0 \end{bmatrix}$$....
1
vote
1
answer
321
views
Action of square root of tridiagonal matrix product on vector
Assume nonsymmetric, tridiagonal matrices $A, B \in \mathbb{R}^{n\times n}$ (where $n$ is in the order of 1000) and $A, B, AB$ are diagonalizable and have positive eigenvalues.
How do you efficiently ...
- 213
1
vote
1
answer
372
views
Relation between the algebraic multiplicity of an eigenvalue and the subdiagonal elements of a symmetric tridiagonal matrix [closed]
Show that if $T$ is a symmetric tridiagonal matrix and an eigenvalue $\lambda$ has multiplicity $k$, then at least $k−1$ subdiagonal elements of $T$ are zero.
If we consider a submatrix $B$ that has ...
2
votes
0
answers
413
views
On a recurrence relation with non-constant coefficients
I am studying a real symmetric tridiagonal matrix $J_{N+1}$ (all off-diagonal elements non-zero) of dimension $N+1$, and I would like to solve the eigenvalue problem. The point is that the ...
- 135
2
votes
1
answer
628
views
Upper Bounds on the Largest Eigenvalue of Jacobi Matrices
Suppose I have a symmetric tridiagonal (Jacobi) matrix in the following form:
$ \begin{pmatrix}
1 & a_{1} & 0 & ... & 0 \\\
a_{1} & 1 & a_{2} & & ... \\\
0 & a_{...
- 137
1
vote
1
answer
2k
views
Relationship between eigenvalues of summation of two matrices one is diagonal
I wonder if someone can prove/disprove the following inequality,
$\lambda_i(A+mI) \leq \lambda_i(A+K) \leq \lambda_i(A+MI)$
where $A$ is a real symmetric Metzler matrix with real and nonpositive ...
- 11
5
votes
1
answer
8k
views
Eigenvalues and eigenvectors of tridiagonal matrices
What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
&...
- 73
11
votes
2
answers
524
views
Matrix exponential, containing a thermal state
This question was originally posted on Mathematics SE, and I'm cross posting it here.
Define an infinite matrix $$ M =
\begin{bmatrix}
0 & -1 & 0 & 0 & \cdots \\
1 & 0 & -2 &...
- 996
8
votes
1
answer
2k
views
Matrix elements of exponential of tridiagonal matrices
Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?
Motivation: I'm trying to find the first passage time ...
- 371
7
votes
1
answer
3k
views
How can I calculate eigenvalues of a tridiagonal matrix? [closed]
Are there special methods to get exact eigenvalues of a tridiagonal matrix?
3
votes
0
answers
236
views
Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
8
votes
0
answers
662
views
Eigenvalues of leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix
It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix
$$\begin{pmatrix}
0 & n-1 & 0 & \dots & 0 \\\
1 & 0 & n-2 & \dots & 0\\\
0 & ...
- 129
11
votes
1
answer
1k
views
Exact eigenvalues of a specific tridiagonal matrix
I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
- 171
4
votes
1
answer
561
views
Eigenvalues of large tridiagonal matrix
Consider large tridiagonal matrix (where $a$ and $b$ are real numbers):
$$M =
\begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
...
- 285
1
vote
2
answers
1k
views
Computing the eigenvalues of a tridiagonal Toeplitz matrix using inverse iteration
I have a symmetric positive definite matrix $A$ defined by
$$ a_{ij} = \begin{cases} 2 & \text{if} & i = j \\ -1 & \text{if} & |i - j| = 1 \\ 0 & \text{if} & \text{otherwise} ...
- 121
3
votes
2
answers
1k
views
Eigenvectors and eigenvalues of Tridiagonal matrix with varying diagonal elements
is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form :
\begin{pmatrix}
1 & b & 0 & ... & 0 \\\
b & 2 &...
- 31
22
votes
6
answers
45k
views
Eigenvalues of symmetric tridiagonal matrices
Suppose I have the symmetric tridiagonal matrix:
$$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & \ddots & \vdots \\\
0 & b_{2} & a & \...
15
votes
4
answers
7k
views
Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix
Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...
- 153
6
votes
1
answer
863
views
Dominant eigenvector of a real symmetric tridiagonal matrix
What is the most efficient way to calculate the dominant eigenvector of a real symmetric tridiagonal matrix? What's the corresponding time complexity bound?
Could someone give me a reference for ...
- 61