All Questions
Tagged with matrices sp.spectral-theory
77 questions
0
votes
1
answer
45
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Expectation of spectral norm of a diagonal stochastic matrix
There is a diagonal matrix $G(t)\in R^{M\times M}$ where the diagonal elements are independent Bernoulli stochastic variables, satisfying $\mathbb{E}(g_i(t) = 1) = b$, and $ \mathbb{E}(g_i(t) = 0) = 1 ...
3
votes
2
answers
392
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
0
votes
1
answer
184
views
Spectrum of a product of a symmetric positive definite matrix and a positive definite operator
Let $\mathbf H$ be an infinite dimensional Hilbert space.
I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
- 63
1
vote
0
answers
62
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What do you call this class of matrices with a unique positive eigenvalue associated to a graph?
I am looking for the name of a class of symmetric matrices $M\in\Bbb R^{n\times n}$ that I can associate to a (finite simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and that have the following ...
- 13.6k
4
votes
0
answers
189
views
Spectrum of large Hilbert matrices
Let $x_k>0$ be a increasing sequence of real numbers, such that
$$\sum_0^\infty\frac1{x_k}<+\infty.$$
Let us form the (infinite) Hilbert matrix $A\in{\bf Sym}({\mathbb N};{\mathbb R})$ with
$$a_{...
- 52.3k
7
votes
0
answers
195
views
Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.5k
2
votes
1
answer
456
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On a theorem of Carlson on the necessary and sufficient condition for a matrix to have $m$ real eigenvalues
Background: In the physics of open quantum systems the Lindbladian $\mathcal{L}$ governs the evolution of quantum states through the Lindblad master equation.
The Lindblad operator usually has ...
- 1,054
1
vote
0
answers
45
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Approximating spectra of (finite rank pertubations of) Laurent operators by spectra of (pertubations of) periodic finite operators
A tridiagonal matrix is a matrix which only has elements on three diagonals.
So for $\alpha, \beta, \gamma \in \mathbb{C}$ consider the bi-infinite tridiagonal Laurent operator $T$ with $\beta $ on ...
- 1,054
1
vote
1
answer
209
views
Eigenvalues invariant under 90° rotation
Consider $N \times N$ matrices
$$A = \begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & 0 & & 0 \\
\vdots & 1 & 0 & \...
- 536
4
votes
1
answer
147
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prove spectral equivalence bounds for inverse fractional power of matrices
The question is an extention to the answered question prove spectral equivalence bounds for fractional power of matrices.
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and ...
- 75
3
votes
1
answer
80
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prove spectral equivalence bounds for fractional power of matrices
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds
$$ c^- x^\top D x \le x^\top A ...
- 75
2
votes
2
answers
264
views
Prove spectral equivalence of matrices
Let $A,D \in \mathbb{R}^{n\times n}$ be two positive definite matrices given by
$$
D =
\begin{bmatrix}
1 & -1 & 0 & 0 & \dots & 0\\
-1 & 2 & -1 & 0 & \dots & 0\\...
- 31
2
votes
0
answers
538
views
Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices
recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
- 21
2
votes
0
answers
121
views
Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
16
votes
3
answers
2k
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Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
- 73
1
vote
1
answer
241
views
Monotonicity of eigenvalues II
In a previous question here, I asked the question below for block matrices and received an answer showing the question is true if $\mathcal B$ is hermitian and false, in general if $\mathcal B$ is non-...
- 536
6
votes
1
answer
601
views
Monotonicity of eigenvalues
We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$...
- 536
5
votes
0
answers
208
views
Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
2
votes
1
answer
178
views
Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them
I'm looking for an elegant way to show the following claim.
Claim: Let $m_1, m_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are ...
- 45
1
vote
0
answers
158
views
Relation between the spectrum of 𝐷𝐴 and 𝐴 where 𝐴 is a bounded linear self-adjoint positive operator and 𝐷 is a constant diagonal positive matrix
Let $D$ be a $3\times 3 $ constant real positive-definite diagonal matrix and $\Omega\subset\mathbb{R}^3$ be a bounded lipschitz domain.
Denote by $(L^2(\Omega))^3$ the set of square integrable ...
- 63
0
votes
1
answer
102
views
Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces [closed]
Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded ...
- 63
11
votes
1
answer
927
views
Imaginary eigenvalues
Consider the matrix
$$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$
This matrix is ...
- 124
13
votes
3
answers
2k
views
Eigenvalue pattern
We consider a matrix
$$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$
One easily ...
- 133
3
votes
1
answer
151
views
Commutation between integrating and taking the minimal eigenvalue
Let $S = (f_{ij})_{ij}$ be a $n \times n$ real symmetric matrix, with functions $f_{ij} \in L^1(\mathbb{R}^d,\mathbb{R})$ in it. We define $\left(\int u S \right)_{ij} = \int u S_{ij}$ as the ...
user140442
3
votes
2
answers
320
views
A relation between norm and spectral radius for some matrix operators on Banach spaces $\ell^{p}$
Let $A=(a_{i,j})_{i,j=1}^{\infty}$ be a semi-infinite matrix with real entries. Suppose further that $A$ and $A^{T}$ (matrix transpose) represent bounded operators on $\ell^{p}$ for $p\geq1$. Denote ...
- 2,188
16
votes
2
answers
1k
views
Spectral symmetry of a certain structured matrix
I have a matrix
$$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$
As ...
- 536
5
votes
3
answers
273
views
Significance of the length of the Perron eigenvector
Let $A$ be a positive square matrix. Perron-Frobenius theory says that there exist $\lambda,v$ with $Av=\lambda v$ and $\lambda$ equals the spectral radius of $A$, $\lambda$ is simple, and $v$ is ...
- 175
9
votes
0
answers
802
views
Positive definiteness of matrix
This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows:
We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
- 192
6
votes
1
answer
299
views
Phase transition in matrix
Playing around with Matlab I noticed something very peculiar:
Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by
$$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$
...
- 536
1
vote
1
answer
2k
views
Positive matrix and diagonally dominant
There is a well-known criterion to check whether a matrix is positive definite which asks to check that a matrix $A$ is
a) hermitian
b) has only positive diagonal entries and
c) is diagonally ...
- 183
4
votes
2
answers
1k
views
What's the full assumption for Laplacian matrix $L=BB^T=\Delta-A$?
Graph with no-selfloop, no-multi-edges, unweighted.
directed
For directed graph Adjacency matrix is a non-symmetric matrix $A_{in}$ considering indegree or $A_{out}$ considering outdegree. Degree ...
- 211
0
votes
1
answer
262
views
Perturbing a normal matrix
Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...
user136156
5
votes
1
answer
417
views
Stable matrices and their spectra
I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices.
A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...
user136156
3
votes
1
answer
791
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
3
votes
1
answer
463
views
Spectrum of this block matrix
Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
- 536
6
votes
1
answer
487
views
Intuitive proof of Golden-Thompson inequality
Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality:
For any hermitian matrices $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...
- 163
0
votes
0
answers
107
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)...
- 16.2k
4
votes
0
answers
148
views
A Toeplitz variant of the Hilbert matrix
It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries
$$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$
determines a bounded operator on $\ell^{2}(\mathbb{N}...
- 2,188
3
votes
0
answers
182
views
Diagonalization of Hermitian Matrix Polynomials
I have a question on the decomposition of polynomial matrices.
Suppose $A(\lambda) = \sum_{j=0}^L \lambda^j A_j$ is an $n \times n$ matrix of polynomials, which is Hermitian on the real axis $\lambda ...
- 31
5
votes
0
answers
262
views
p-adic analogue of self-adjoint operator
Consider the very well-known result that any Hermitian matrix over $\mathbb{C}$, say $T$, admits a decomposition $T = UDU^*$ where $U$ is unitary and $D$ is diagonal with real entries. I am looking ...
2
votes
1
answer
968
views
Eigenvectors of symmetric positive semidefinite matrices as measurable functions
I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...
1
vote
1
answer
2k
views
Largest element in inverse of a positive definite symmetric matrix [closed]
If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
- 15
1
vote
1
answer
307
views
Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?
I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using,
H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\...
- 15
6
votes
0
answers
587
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
- 907
4
votes
2
answers
477
views
Non-asympototic version of Gelfand's formula
Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...
- 181
0
votes
0
answers
160
views
$l_{\infty}$ norms of matrix perturbations
Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that $\lambda_{max}(...
- 1,893
1
vote
0
answers
455
views
Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition
Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
- 267
2
votes
1
answer
86
views
Information on special matrices similar to Jacobi matrices
Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ...
- 2,188
3
votes
0
answers
362
views
Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix
$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n} $ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where $\...
- 143
7
votes
0
answers
217
views
Characterizing matrices with rank constraint
Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...
- 13.9k