# 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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### 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 &...
217 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 ...
356 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 ...
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1 vote
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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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### 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
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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 ...
1 vote
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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 ...
281 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 ...
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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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