All Questions
Tagged with fa.functional-analysis matrix-analysis
65 questions
0
votes
0
answers
510
views
Composition of upper semi-continuous real valued function with upper semi-continuous matrix valued function
Say that a matrix valued function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is upper semi-continuous at $x_0$ if
$$ \limsup_{x \rightarrow x_0} A(x) \preceq A(x_0), $$
where $\preceq$ ...
- 1,590
6
votes
1
answer
2k
views
Comparing norms on tensor products of matrices
Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is
$$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$
where $(s_1,...
- 209
13
votes
1
answer
1k
views
A generalization of the Powers-Stormer inequality
The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...
- 2,019
3
votes
2
answers
939
views
Positive definiteness of infinite tridiagonal matrices
I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \...
- 692
6
votes
1
answer
277
views
Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$
Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...
- 109k
6
votes
0
answers
243
views
Operator arithmetic-harmonic mean inequality with operator-valued weights
Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
- 4,000
2
votes
1
answer
346
views
Alike looking matrices imply convergence of eigenvalues?
This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and ...
user54300
3
votes
0
answers
193
views
Method to Generate Random Mutually Orthogonal Unitary Matrices
The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
- 133
15
votes
1
answer
889
views
Operator norms of circulant matrices
The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.
For complex numbers $a_1,\ldots,a_n$, I will use the notation
$$
\mbox{...
- 717
8
votes
1
answer
1k
views
Is the p-norm of a matrix strictly log-convex?
Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is,
$$
\|A\|_p = \sup_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_p}...
- 3,497
2
votes
1
answer
276
views
Asymptotic Behavior of Non-Analytic Function of the Eigenvalues
Hello,
Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$.
If $A_n$ were a sequence of Hermitian ...
- 43
3
votes
2
answers
171
views
a monotone relation for s-numbers
Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one $s_n(...
- 478
8
votes
4
answers
13k
views
Eigenvalues of infinite matrices [closed]
I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can ...
- 431
7
votes
0
answers
355
views
An $L^{\infty}$ version of principal component analysis?
I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal.
I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
- 193
12
votes
2
answers
1k
views
Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?
Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...
- 123