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2
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0answers
19 views

Approximate Simultaneous Diagonalization of Non-Hermitian Matrices

Let $A_1,A_2$ two $n\times n$ complex matrices. $A_1$ and $A_2$ are also non-normal, especially, non-hermitian and do not commute. I would like to find an invertible matrix $V$ such that $$ \sum_{i=1,...
4
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0answers
43 views

Lichnerowicz-Obata theorem for symmetric 2-tensor

Let $(S^n/\Gamma,g)$ be the standard space-form with constant sectional curvature, where $\Gamma \subset O(n+1)$ is a finite group. If there exists a nonzero transverse-traceless symmetric 2-tensor $h$...
5
votes
1answer
121 views

Bounding the eigenvalues of $B A B^T$ with the eigenvalues of $A$

Given a Hermitian positive semi-definite $n \times n$ matrix $A$ and a rectangular $m \times n$ matrix $B$, is there anything that can be said about the eigenvalues of the matrix $B A B^T$? It seems ...
0
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0answers
65 views

Expected value of eigenvalue of matrix

Let $A = (X_{ij})_{ij}$ a square matrix of size $n$ where the $X_{ij}$ are (discrete) real random non-negative entries. Denote by $\lambda_1(A) \geq \dots \geq \lambda_n(A)$ the (random) ordered ...
1
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0answers
164 views

spectral property of irreducible matrices

We have an irreducible non-negative matrix $B$ in which all the diagonal elements are zero and the other entries can be 0 or 1. Moreover, $B$ is diagonalizable. We partition the rows of this matrix to ...
1
vote
1answer
50 views

Efficient way to compute eigenvalue decomposition for following problem

I have an optimization problem $$\begin{array}{ll} \text{minimize} & Tr(X^TAX) \\ \text{subject to} & X^TX=I \end{array}$$ where $A\in R^{n \times n}$ and it is symmetric positive definite, ...
3
votes
2answers
96 views

Eigenvalue density of a symmetric tridiagonal matrix

Let $A_n\in\mathbb{R}^{n\times n}$ be defined as $$ A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
0
votes
1answer
32 views

Will truncated SVD ever flip the sign of any element of the matrix?

For a symmetric p.s.d matrix $A \in \mathcal{R}^{n\times n}$, we can calculate its SVD as $A=USV^T$, then we can use the truncated SVD to approximate it with a low-rank matrix $\tilde{A} = \sum_i^...
2
votes
1answer
76 views

Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
0
votes
1answer
76 views

How eigenvalue perturbation affects back to the original matrix?

Both "eigenvalue perturbation" and "matrix perturbation" seems to study this scenario that given a matrix $A$, if we add something to it like $\tilde{A} = A+E$, how will the eigenvalues of $\tilde{A}$ ...
1
vote
1answer
50 views

On the eigenvalue of the expectation value of a random matrix in quadratic form

When we handle with some dynamic input-output mappings, there occurs a question as follows: Let $M$ be a random matrix, of which each element contains random terms. Consider the two expectation ...
2
votes
1answer
99 views

Eigenvalues of A^T D A for positive A and diagonal D

Suppose I have a diagonal matrix $D$ whose entries are bounded in absolute value. I also have a matrix $A$ that is positive (entry-wise, so $A_{ij} > 0\ \forall\ i,j$): one can assume that the ...
1
vote
1answer
89 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_{...
1
vote
1answer
32 views

Asymptotic eigenvalue distribution of sum of two i.i.d random matrices with Marchenko Pastur distributed eigenvalues?

Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where: $\...
0
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0answers
21 views

Found a bound which is better than spectral variation bound under some special case

It is well know that if $\lambda, \mu >0$ are the two leading eigenvalues of two $s \times s$ matrices $A$ and $B$ respectively then \begin{align} \lambda - \mu \leq \left(\|A\| + \|B\|\right)^{1-...
2
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0answers
97 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
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2answers
164 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
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0answers
31 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
48 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 ...
2
votes
0answers
56 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
74 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
133 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
208 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
126 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
55 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 ...
9
votes
0answers
155 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
105 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
131 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
393 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
81 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
100 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
42 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
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0answers
111 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
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0answers
25 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
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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
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0answers
41 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
96 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
203 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
129 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
98 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
69 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
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0answers
54 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
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0answers
158 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
115 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
128 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
137 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
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0answers
111 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
159 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
136 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)...