# 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.

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### 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 & &\\
& &...

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### 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 & \...

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### 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 & \...

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### 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 &...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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 ...

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### 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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### Eigenvalue distribution of a band 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 ...

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### 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 ...

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1
answer

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### 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
$$B=\begin{bmatrix}0&{A} \\{A}^T &0 \end{bmatrix}$$....

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1
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### 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 ...

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1
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### 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 ...

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### 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 ...

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1
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### 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_{...

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1
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### 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 ...

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1
answer

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### 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 \\
&...

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votes

1
answer

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### 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 ...

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### How can I calculate eigenvalues of a tridiagonal matrix? [closed]

Are there special methods to get exact eigenvalues of a tridiagonal matrix?

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### 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 &...

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### 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 & ...

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1
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### 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 & ...

4
votes

1
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### 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 & \\
...

18
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5
answers

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### Eigenvalues of Symmetric Tridiagonal Matrices

Suppose I have the symmetric tridiagonal matrix:
$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & & ... \\\
0 & b_{2} & a & ... & 0 \...

13
votes

4
answers

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### 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 &...

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### 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 ...