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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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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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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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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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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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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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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3 votes
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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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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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7 votes
1 answer
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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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3 votes
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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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9 votes
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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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11 votes
1 answer
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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 & ...
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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 & \\ ...
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15 votes
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 \...
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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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6 votes
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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 ...
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