The eigenvalues tag has no usage guidance.

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### 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

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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) ...

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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}{\...

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votes

**1**answer

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 ...

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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 $...

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votes

**1**answer

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=[...

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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 ...

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votes

**1**answer

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 \\
...

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**2**answers

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 ...

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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 ...

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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
...

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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, ...

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**1**answer

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 ...

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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, ...

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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 $...

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**1**answer

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 $A$?

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**1**answer

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 \...

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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.
...

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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 $m$ ...

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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 - ...

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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 \...

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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{...

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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)-(...

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**1**answer

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 ...

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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) = \...

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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 ...

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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)...

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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^...

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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 ...

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**1**answer

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 $R_s = U_s\Lambda_sU_s^*$. ...

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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-...

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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 ...

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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}\...

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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 ...

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votes

**1**answer

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)...

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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 ...

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**1**answer

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 ...

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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 ...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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}^...

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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 ...

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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 ...

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**1**answer

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 $G$ ...

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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&...

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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\...

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**1**answer

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 ...

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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 \...