Tagged Questions

0
votes
0answers
18 views

Spectral Properties of Concatenated Matrices

I am wondering if anyone is aware any resource on the internet that systematically studies the spectral properties of submatrices/ concatenated matrices. I am interested in proving/ disproving the ...
4
votes
0answers
31 views

Gelfand's formula, different field

Gelfand's formula says that for a complex matrix $A \in \mathbb{C}^{n \times n}$, $$\rho(A) = \lim_{m \rightarrow \infty} \|A^m\|^{1/m},$$ where $\rho$ is the spectral radius (norm of maximal ...
2
votes
1answer
49 views

spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
0
votes
1answer
36 views

real spectrum of an almost symmetric stochastic matrix

Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each ...
0
votes
2answers
66 views

Are any matrices positive semidefinite, non-negative, and not diagonally dominated?

If so, I'd appreciate any examples. Thanks.
2
votes
2answers
79 views

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix ...
0
votes
0answers
82 views

Eigenvalues of weighted Laplacian matrix $L$ and $ML$, where $M$ is a diagonal matrix with positive entries

I have a weighted Laplacian matrix of a directed graph and a diagonal matrix $M$ with positive entries. Is it possible to establish a relation between the eigenvalues of $L$ and those of the product ...
0
votes
0answers
50 views

Fastest way to compute spectral radius

Let's say that I have calculated the solution of system of linear equations using Jacobi or Gauss-Seidel iterative method. Using the previous calculation results, what is the fastest way to compute ...
2
votes
0answers
64 views

$L^p$ norm of diagonal is $\leq$ Schatten $L^p$ norm

Let $A = (a_{ij})$ be an $n\times n$ symmetric matrix. Its Schatten norm is defined by $\|A\|_{S^p}^p = \sum_{j=1}^n |\lambda_j|^p$, where $\lambda_j$ are the eigenvalutes of $A$. I am trying to prove ...
2
votes
1answer
145 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
2
votes
1answer
119 views

Decompose a complex symmetric matrix to retain positive definitness

I have a complex symmetric matrix $A$, (i.e. non-Hermitian and obeying $A=A^T$), which is positive definite, in the sense that: $$\Re({z^HAz}) > 0$$ for any $z$. I am able to verify this ...
9
votes
1answer
510 views

How to understand spectral decomposition geometrically

Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have $$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$ and $$A^{-1} = ...
1
vote
0answers
67 views

Spectrum of Transition Matrix for Random Walk

Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
1
vote
1answer
129 views

operator norm and spectral radius

is it true that the operator norm of a matrix $A$ is smaller than 1 if its spectral radius $\rho(A)$ is smaller than 1? many thanks for any help, it is much appreciated!
2
votes
0answers
169 views

Spectral radius and positive definite of matrices

Denote $ \rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
1
vote
1answer
154 views

Similar matrix proof

$A$ and $B$ are similar matrices, if $B=PAP^{-1}$ holds for a square, non-singular matrix $P$. Now am wondering if $S^{-1}T$ and $S^{-1/2}TS^{-1/2}$ are similar matrices? Am looking for a proof for it ...
2
votes
0answers
47 views

Root Convergence rate of Iterative Scheme [closed]

I have an iterative sequence for optimizing an EM algorithm based loss function $L(X)$ with $t$ being the iteration number as: $X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is a diagonal matrix, $B$ and ...
3
votes
1answer
196 views

are there any bounds on the eigenvalues of products of positive-semidefinite matrices?

I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$. I am looking for upper bounds and lower bounds on the $m$-th largest eigenvalue of $AB$, in terms of the ...
0
votes
1answer
90 views

The row- and column-sums of a nonengative matrix with spectral radius less than $1$

Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$? Edit: What if the matrix has positive ...
3
votes
1answer
111 views

Behavior of the spectral radius of a convergent matrix when some of the elements of the matrix change sign

I want to prove (or disprove) the following statement: If $A$ is a square matrix with non-negative elements that has spectral radius less then $1$, then any matrix obtained from $A$ by ...
1
vote
1answer
181 views

How to solve X*A=C matrix equation where two (X and A) matrices are unknown?

I have a spectroscopy problem that boils down to a matrix equation where X*A=C. I take N observations each consisting of 3 detector readings and my detectors suffer from some amount of cross-talk ...
16
votes
1answer
523 views

Ratio of largest eigenvalue to sum of eigenvalues — where to read about it?

Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio $$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$ is a measure of the "rank-one-ness" ...
4
votes
2answers
395 views

Matrix Differential Equation with a Skew-Symmetric Matrix

From a bank of masters exams: Say the position of a particle moving in $\mathbb{R}^n$ is given by a smooth vector-valued function $\vec{x}(t)$. Suppose that $\vec{x}(t)$ satisfies a ...