All Questions
Tagged with sp.spectral-theory matrices
77 questions
29
votes
3
answers
3k
views
Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 2,200
20
votes
2
answers
8k
views
Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters
Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
- 4,312
17
votes
5
answers
2k
views
Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?
This question is related to another question, but it is definitely not the same.
Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
- 4,312
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
16
votes
3
answers
2k
views
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
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
12
votes
6
answers
692
views
Invertibility of a certain matrix indexed by the Hamming cube
For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim
On submeasures on Boolean algebras, arXiv 1212.6822v3
and in Section 7 the ...
- 25.8k
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
9
votes
1
answer
1k
views
0 eigenvalue for a symmetric tridiagonal matrix
Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...
- 143
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
8
votes
0
answers
738
views
Bounding sum of first singular values squared for Kronecker sum of traceless matrices
Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
- 1,612
7
votes
1
answer
1k
views
Origins of the Jacobi matrix
I have several questions concerning history of Jacobi matrices.
Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...
- 2,188
7
votes
1
answer
271
views
Singular values of $X+iY$ where $X$ and $Y$ are Hermitian
Lets have two Hermitian $n\times n$ matrices $X$ and $Y$.
Are there any known properties of the singular values of
$$Z = X + i Y.$$
I am the most interested in bounding from above a few first ...
- 1,612
7
votes
0
answers
195
views
Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.5k
7
votes
0
answers
217
views
Characterizing matrices with rank constraint
Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\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
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
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
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 $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...
- 163
6
votes
0
answers
587
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
- 907
5
votes
2
answers
1k
views
spectral radius monotonicity
I encountered an inequality when reading a paper. Can someone help to show how to prove it?
Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
- 2,239
5
votes
3
answers
273
views
Significance of the length of the Perron eigenvector
Let $A$ be a positive square matrix. Perron-Frobenius theory says that there exist $\lambda,v$ with $Av=\lambda v$ and $\lambda$ equals the spectral radius of $A$, $\lambda$ is simple, and $v$ is ...
- 175
5
votes
1
answer
600
views
Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle....
- 4,188
5
votes
1
answer
417
views
Stable matrices and their spectra
I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices.
A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...
user136156
5
votes
3
answers
271
views
Conditions ensuring an order betweenthe smallest eigenvalues of two positive definite Jacobi matrices
Let $J, L$ be two symmetric positive definite tridiagonal matrices of positive diagonal entries, $\mbox{diag}(J)=(a_1, a_2, \ldots, a_n)$, $\mbox{diag}(L)=(\alpha_1, \alpha_2, \ldots, \alpha_n)$, ...
- 143
5
votes
0
answers
208
views
Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
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 ...
5
votes
0
answers
539
views
An inverse eigenvalue problem on Jacobi matrices
I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
- 291
4
votes
5
answers
4k
views
About adding a negative definite rank-1 matrix to a symmetric matrix
If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...
- 1,893
4
votes
1
answer
147
views
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
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
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
4
votes
1
answer
1k
views
dominant eigenvector
Hi, everyone! Is there any efficient way to simplify the following tensor product
$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.
My goal is to efficiently compute the ...
- 41
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
$$a_{...
- 52.3k
4
votes
0
answers
148
views
A Toeplitz variant of the Hilbert matrix
It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries
$$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$
determines a bounded operator on $\ell^{2}(\mathbb{N}...
- 2,188
3
votes
2
answers
392
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
3
votes
3
answers
3k
views
Infinite hermitian matrix
Suppose we have a finite square n x n matrix of complex numbers H that is Hermitian and skew-symmetric:
$H^\dagger = H$ and $H^T = -H$.
(T denotes transpose, $\dagger$ denote conjugate transpose. I ...
- 195
3
votes
1
answer
284
views
Estimating spectral radius with a Gaussian vector
Suppose I'm trying to estimate the spectral radius of a square $n \times n$ matrix $A$,
and let $N$ be a distribution over Gaussian i.i.d. vectors of length $n$.
Is the following lemma true:
If the ...
- 33
3
votes
1
answer
463
views
Spectrum of this block matrix
Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
- 536
3
votes
1
answer
944
views
numerical range of a column-zero-sum matrix
I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...
- 3,388
3
votes
1
answer
80
views
prove spectral equivalence bounds for fractional power of matrices
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds
$$ c^- x^\top D x \le x^\top A ...
- 75
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
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
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
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:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
2
votes
1
answer
456
views
On a theorem of Carlson on the necessary and sufficient condition for a matrix to have $m$ real eigenvalues
Background: In the physics of open quantum systems the Lindbladian $\mathcal{L}$ governs the evolution of quantum states through the Lindblad master equation.
The Lindblad operator usually has ...
- 1,054
2
votes
1
answer
968
views
Eigenvectors of symmetric positive semidefinite matrices as measurable functions
I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...
2
votes
1
answer
86
views
Information on special matrices similar to Jacobi matrices
Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ...
- 2,188
2
votes
2
answers
264
views
Prove spectral equivalence of matrices
Let $A,D \in \mathbb{R}^{n\times n}$ be two positive definite matrices given by
$$
D =
\begin{bmatrix}
1 & -1 & 0 & 0 & \dots & 0\\
-1 & 2 & -1 & 0 & \dots & 0\\...
- 31
2
votes
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