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2
votes
0answers
65 views

eigenvalues of a square block matrix

How can we show that there are not defective eigenvalues for this square block matrix of dimension $2d \times 2d $: \begin{bmatrix} A&B\\-B& 0 \end{bmatrix} where A, B are real matrices, $A =\...
3
votes
2answers
153 views

Find parameter values for which a 3x3 matrix has a triple eigenvalue

An Exceptional point generally occurs in eigenvalue problems in which the matrix is dependent on some parameter(s). The particular point in which the eigenvalues become degenerate for the parameter(s) ...
0
votes
0answers
28 views

Eigenvalue sensitivity matrix calculation for perturbation.

I have a matrix $J(x)=J_o+J_d(\Delta x)$. I got the expression of eigenvalue sensitivity matrix by partial differentiating the relation of eigenvalue and eigenvectors: $$\frac {\partial\lambda}{\...
0
votes
1answer
43 views

Bounding/approximating the largest eigenvalue of the special case of companion matrix

Suppose I have the following companion matrix ($d\times d$) The companion matrix A. $1 \geq p \geq q \geq 0$. Let $x$ ($d\times 1$) be the all one vector and my underlying problem is to analyze the ...
1
vote
0answers
47 views

Computational complexity for spectral radius of symmetric matrix

What is the best known algorithmic complexity for computing the spectral radius (largest eigenvalue in magnitude, possibly with respect to some precision and confidence) of a symmetric matrix of size $...
2
votes
1answer
61 views

What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?

I have two problems related to eigenvalues of negative definite matrices: I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M_{11}-M_{12}M_{22}^{-1}M_{21}$ by taking $M=[...
3
votes
0answers
118 views

Eigenvectors of sum of SO(3) matrices

I asked this question before on MSE but go no answers. It seems that the problem is rather difficult so I thought of trying here. Given two matrices $A,B\in SO(n)$, each describing a rotation by ...
5
votes
1answer
184 views

Largest Eigenvalue of a Matrix with Special Form in terms of n

In one step of solving a difficult problem, I would like to know the largest eigenvalue of a matrix with this pattern: $$A_n = \begin{bmatrix} 0 & 0 & 0 & 0 &\dots & 0 \\ ...
3
votes
2answers
111 views

Upper bound of spectral radius of the sum of two matrices, one with spectral radius no larger than 1, and the other has small eigenvalues

Suppose I have one $pN\times pN$ matrix $\bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $\bf H$ is in a block-like ...
1
vote
0answers
49 views

Upper bounds on absolute eigenvalue of sum of two matrix

We have this iteration $$X_{k+1}=(G\cdot Jf+H)X_k+C$$ with $G$ is symmetric and nonnegative, $H$ is nonnegative. $Jf$ is the jacobian matrix of some function $f$ and we can assume it satisfy certain ...
8
votes
0answers
147 views

Maximum dimension of a space of $n\times n$ real matrices with at least $k$ nonzero eigenvalues

Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every ...
2
votes
0answers
74 views

About product of PSD matrices

In Theorem 3 in this paper, https://core.ac.uk/download/pdf/82822897.pdf, ``On a product of positive semidefinite matrices, A.R. Meenakshi, C. Rajian, Linear Algebra and its Applications, Volume 295, ...
4
votes
1answer
127 views

How to find the analytical representation of eigenvalues of the matrix $G$?

I have the following matrix arising when I tried to discretize the Green function, now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
16
votes
2answers
370 views

Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$

Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, ...
1
vote
1answer
67 views

Lanczos algorithm for finding $k$ smallest eigenvector

I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
-2
votes
1answer
95 views

What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]

Suppose we have the following symmetric matrix. $$A = \sigma^2 I + u u^T$$ What can we say about the eigendecomposition of $A$?
0
votes
1answer
40 views

Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?

Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is: $$ K (t,s) = \sum_{i=1}^\infty \lambda_i \...
1
vote
0answers
102 views

Sufficient conditions for all eigenvalues simple in stochastic matrix

The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical Perron-Frobenius theorem. In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple. ...
1
vote
0answers
22 views

What is the distribution of engenvalues of covariance matrix when the covariance has some block diagonal structure

Let's say we have a matrix $X \in \mathbb R^{n\times p}$, where $X_{i,j}$ sampled from a Gaussian $N(\mu, \sigma^2)$, we use $\Phi$ to denote $\{\mu,\sigma\}$ for simplicity. Now, we sample $m$ ...
1
vote
0answers
35 views

Change of variables between quadrilaterals - Rayleigh quotient

A - Vertex at bottom left B - Vertex at bottom right K - Vertex at top left of blue quadrilateral C - vertex at top left of brown quadrilateral L - vertex at top right of blue quadrilateral F - ...
0
votes
0answers
40 views

The eigenvalues $\lambda$ of $\lambda \phi_j(x)= \int_G{ K(x-y)\phi_j(y)dy}$

Prove the eigenvalues $\lambda$ of $\lambda \phi_j(x)= \int_G{ K(x-y)\phi_j(y)dy}$ is $\int_G{K(x)\phi_{-j}(x)dx}$, with $\phi_j(x)=(2R)^{-n/2}exp(i\pi j. \frac{x}{R}), j \in \mathbb{Z}^n, x, y \in \...
3
votes
1answer
93 views

Inertial decomposition of graphs

The problem is this: given a graph $G$, to find a decomposition of $G$, i.e. a set $F$ of vertex-disjoint proper subgraphs of $G$ such that: $$\text{inertia}(G) = \sum_{H \in F} \operatorname{...
1
vote
1answer
200 views

Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset $$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$ is a manifold of dimension $2n(2r)-(...
2
votes
1answer
127 views

When does a row standardized adjacency matrix have a real spectrum?

A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions ...
1
vote
0answers
94 views

Primes approximated by eigenvalues?

Let the matrix $T$ be defined by: $$\displaystyle T(n,k) = -\varphi^{-1}(\operatorname{GCD}(n,k))$$ where $\varphi^{-1}$ is the Dirichlet inverse of the Euler totient function. $$\varphi^{-1}(n) = \...
0
votes
1answer
68 views

How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$ Find the marginal distribution of each eigenvalue, using whatever you can. Background In my ...
0
votes
0answers
52 views

Numerical error on the spectrum of a matrix

Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
4
votes
0answers
155 views

How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A? ‎ $$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^...
0
votes
0answers
65 views

necessary and sufficient condition for purely imaginary generalized eigenvalues

Consider the generalized eigenvalue equation $$A \mathbf{v}=\lambda S \mathbf{v}$$ where $S$ is a real square symmetric matrix and $A$ a real square anti-symmetric matrix. I seek a necessary and ...
2
votes
1answer
114 views

Connections between eigenvectors after matrix multiplication

Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...
0
votes
0answers
124 views

Smallest eigenvalue of a sparse matrix (updated)

Let $D_{1}$ be $(m-1)n \times mn$ matrix (that is, $(m-1)n$ rows and $mn$ columns) and $D_{2}$ be $m(n-1) \times mn$ defined as $$\begin{cases} D_{1}[(m-1)(j-1)+i ; m(j-1)+i] & = -1 , \\ D_{1}[(m-...
4
votes
1answer
134 views

when is an eigenvalue differentiable with respect to a parameter? [duplicate]

Let say we have a symmetric matrix $A(\omega)$ depending smoothly on some variables $\omega \in \Omega$ with $\Omega \subset \mathbb{R}^d$ a $d$-dimensional parameterspace (this means the eigenvalues ...
2
votes
0answers
107 views

Relationship between eigenvalues of compact operators $A$ and $(A+A^*)/2$

A result from 'Topics in Matrix Analysis' by Horn & Johnson (3.3.33) is the following: For $A\in \mathbb{M}_n$, $\sum_{i=1}^k Re \lambda_i(A) \leq \sum_{i=1}^k Re \lambda_i \big(\frac{A+A^*}{2}\...
1
vote
1answer
148 views

Eigenvalue Argument Perturbation

Given two square matrices $A$ and $B$. There are quite some results on the distance between the eigenvalues, e.g., $$ | \lambda_A - \lambda_B | \leq \| A - B \|_F, $$ where $A$ and $B$ are Hermitian ...
3
votes
1answer
111 views

Distribution of eigenvalues of a Wishart matrix

Is there a known expression for the eigenvalue distribution of a matrix of the form $$\sum\limits_{i=1}^n k_ia_ia_i^T$$ where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)...
0
votes
0answers
45 views

Change of polynomial eigenvalues between polynomials

Given the polynomial eigenvalue problem $$ p_t(z) = det ( P(z) + Q(t) ) = 0, $$ where $P(z) = \sum_{i=0}^k P_i z^i$ with $P_i \in \mathbb{C}^{n \times n}$ and $Q(t) \in \mathbb{C}^{n \times n}$. The ...
2
votes
1answer
126 views

Rotatable matrix, its eigenvalues and eigenvectors

We say that a real matrix is rotatable iff after turning it clockwise on $90^{\circ}$ it doesn't change. I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such ...
12
votes
3answers
767 views

Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
4
votes
2answers
115 views

Lower bound on the entries of the Perron vector

Let $A$ be a matrix that satisfies all the conditions of Perron- Frobenius theorem. From the theorem it is known that the entries of the eigenvector corresponding to the largest eigenvalue will be ...
3
votes
0answers
91 views

Maximizing a certain eigenvalue ratio

Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
2
votes
0answers
65 views

Decay rate of least eigenvalue of Gram matrices

Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$: $$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$ In ...
2
votes
0answers
123 views

List of analytically known eigensystems?

In condensed matter physics, we often come across matrices that are multi-diagonal or banded. For example, I may have a matrix with three tridiagonal bands, or a tridiagonal band and two/four ...
0
votes
0answers
21 views

sensitive perturbation approximation

I was reading paper which associated with perturbation approximation. paper1 paper2. In paper1: $\bar{R}=R+\epsilon C$, first order: when $\Lambda_1\gg\Lambda_2$, $\Delta\Lambda_{max}=\frac{\vec{v}^...
2
votes
0answers
115 views

Eigenvalues of special sum of Hermitian matrices

In my research on linear algebra and its applications, I have come across the following problem which has stumped me: Let $ A $ be a positive definite matrix and let $ D $ be a positive diagonal ...
2
votes
0answers
80 views

How to compute the Eigen values of diagonal plus a rank one matrix? [duplicate]

I'm trying to find information on the eigenvalues of an n×n matrix $A$ such that $A=D+J$ Where $D$ is some complex valued diagonal matrix, and $J$ is a rank one matrix, $J = uu^T$. How to compute the ...
3
votes
1answer
148 views

Eigenvalues of a generalized Gram matrix

Let $P \in \rm{GL}_N(\mathbb{C})$. Call the columns of $P$ $|R_i\rangle$ and the rows of $P^{-1}$ $\langle L_i|$, so that $$ \langle L_i | R_j \rangle = \delta_{i,j}. $$ And define the matrix $G$ ...
2
votes
0answers
63 views

Asymptotic behavior of the Dirichlet-Laplacian eigenvalues [closed]

I found in a math book http://www.cambridge.org/dz/academic/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-partial-differential-equations?format=PB&...
0
votes
0answers
15 views

Algorithm for computing capacity of channel with additive colored Gaussian noise

I am looking for an algorrithm or Matlab/Python script for computing the capacity of additive colored Gaussian noise channel, according to the formula: $$C=\frac{1}{n}\sum_{i=1}^{n}{\frac{1}{2}}\log\...
1
vote
1answer
59 views

Polynomial Eigenvalue Problem with few non-zero coefficients

Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...
4
votes
1answer
331 views

The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices? [closed]

Problem: Suppose we have two real, symmetric and positive definite square matrices $A$ and $B$, i.e., $$A_{ij}, B_{ij}\in \mathbb{R}$$ $$A^T=A$$ $$B^T=B$$ $$x^TAx>0 \forall x$$ $$x^TBx>0 \...