All Questions
Tagged with fa.functional-analysis linear-algebra
325 questions
0
votes
0
answers
174
views
Lipschitz map on positive definite cone of $n$-by-$n$ matrices
A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
- 91
8
votes
1
answer
446
views
Parallelogram law for vectors of equal length
Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert ...
- 1,361
4
votes
1
answer
219
views
Is a mixture of $\ell_p$-norms $\eta(x):=\lVert x\rVert_2 + r\lVert x\rVert_p$ always dimensionlessly equivalent to some $\ell_q$-norm?
$\newcommand\norm[1]{\lVert#1\rVert}$For any $p \in [1,2]$, $r \ge 0$, and integer $d \ge 1$, define a mixed-norm $\eta:\mathbb R^d \to \mathbb R$ by $\eta(x) := \norm x_2 + r\norm x_p$, for any $x \...
- 6,853
1
vote
0
answers
87
views
Continuous choice of null directions for a family of bilinear forms
Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
- 2,095
2
votes
0
answers
81
views
An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$
Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define
$$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$
...
- 4,058
2
votes
1
answer
174
views
Gradient of a convex function on $\mathbb{R}^d$, maximum on hypercubes bounded by values in corners?
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex.
For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that
$$
(f'...
- 1,095
0
votes
0
answers
92
views
Positive definite matrix and Hörmander theory
Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$
Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set
$$
\varphi_{\...
- 101
1
vote
1
answer
155
views
Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
- 1,054
2
votes
0
answers
57
views
spilt the sum of singular values of matrices
Let $A_{i} \in GL(d, \mathbb{R})$ for $i=1, 2, 3.$ For $q>0$, we denote $t_{3}^{q}=\sum_{i=0}^{3} \sigma_{1}^{q}(A_{i})\sigma_{2}^{q}(A_{i})\sigma_{3}^{q}(A_{i})$, $t_{2}^{q}=\sum_{i=0}^{3} \sigma_{...
- 1,043
1
vote
1
answer
322
views
A particular commutator of the discrete Fourier matrix
For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
- 4,058
13
votes
2
answers
915
views
Topological vector spaces (reference request)
In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
- 674
1
vote
2
answers
251
views
Linear independence of translates of a function $\{ \phi(\cdot - x) : x \in \mathcal F \}$
Let $0 \neq \phi \in L^2(\mathbb R^n)$ be a square-integrable function and $\mathcal F \subset \mathbb R^n$ a finite set. If we are in the one-dimensional setting $n=1$ then the set of translates of $\...
- 437
2
votes
1
answer
547
views
Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
- 49
3
votes
0
answers
77
views
Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 41
2
votes
2
answers
631
views
Decomposition of a positive definite matrix
Let $K(x)_{n\times n}$ be a positive definite matrix defined on $x\in D$ and $K_{i,j}(x)\in C^2(D)$ (or generally $C^k$) for any $1\le i,j\le n$. Of course for any $x$, there exists a invertable ...
- 379
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
4
votes
2
answers
372
views
How close to uniform are Perron-Frobenius eigenvectors?
Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have ...
- 20.2k
4
votes
0
answers
311
views
Estimates of the Frobenius norm of commutator
Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
- 949
2
votes
1
answer
145
views
Expected matrix created from two random orthogonal-projection matrices
Consider an arbitrary finite set of orthogonal-projection matrices (symmetric, idempotent, etc.) in $\mathbb{R}^{n\times n}$.
We draw two matrices $Q,P$ uniformly and i.i.d. from this set.
Question: ...
- 673
3
votes
0
answers
275
views
Schur-Horn theorem for principal submatrices
The Schur-Horn theorem says that there exists a Hermitian matrix with diagonal entries $d_1,d_2,\ldots,d_n$ and eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$ if and only if $(\lambda_1,\lambda_2,\...
- 6,351
1
vote
0
answers
64
views
Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?
Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed.
Then, they are simultaneously ...
- 3,477
0
votes
0
answers
112
views
How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
- 101
2
votes
0
answers
267
views
Example of a unital contractive map that is not completely positive on an operator system
I am aware of maps that are positive but not completely positive (for example transpose map). BUT I can not think of an example of the following type.
Does there exist an operator $T$ such that a map $...
- 231
3
votes
0
answers
229
views
$f(x)>0$ and $f(y)>0$ implies $f(x+y)>0$, then there must exist an linear function $g$ such that $g(x)>0$ iff $f(x)>0$?
Background: Let $x,y\in\mathbb (0,+\infty)^n$. $f$ is a continuous function on $\mathbb R^n_+=(0,+\infty)^n$. Consider the following condition (1), the sign of $f(x+y)$ is dependent on the sign of $f(...
- 263
3
votes
1
answer
391
views
Factoring higher-order differential operators
I have been researching various methods for solving differential equations. In particular, I want to better understand the factoring approach. For example, if we want to solve a general second order ...
- 51
10
votes
1
answer
337
views
What are the predictive implications of conditional non-commutative probability?
To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$.
In this context a state $S$ is a positive semi-definite ...
- 323
1
vote
1
answer
119
views
Eigenvalues under linear transformation
Let $X$ and $Y$ be square non-symmetric matrices of the same size. Assume that their eigenvalues are close in the sense that there exists a small $\varepsilon>0$ such that, for any eigenvalue $\...
- 31
4
votes
1
answer
204
views
Making Hermitian matrices almost commute
Consider two Hermitian matrices $A, B \in \mathbb{C}^{n \times n}$. I'm interested in finding another Hermitian matrix $A'$ that is close to $A$ and almost commutes with $B$. More precisely, I'd like ...
- 209
0
votes
0
answers
198
views
eigenvalues of the product of a unitary with a diagonal
In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 4,058
1
vote
1
answer
72
views
Weak majorizations for sum of two hermitian matrices
Let $A$ and $B$ be two $n\times n$ hermitian matrices. Does $U^{*}AU+B \prec_{w} A+B$ for any unitary matrix $U$? Here the notation $``\prec_{w}"$ stands for the weak majorization, that is, $x\prec_{...
- 45
7
votes
1
answer
384
views
Bounding the decrease after applying a contraction operator $n$ vs $n+1$ times
Can we upper bound the convergence rate of
$$\max_{\textbf{v}: \left\Vert \textbf{v}\right\Vert_2=1} \left\{ \left\Vert \textbf{T}^n \textbf{v}\right\Vert^2_2 - \left\Vert \textbf{T}^{n+1} \textbf{v}\...
- 673
6
votes
1
answer
299
views
Infinite-dimensional projections of linearly independent sets
A subset of a linear space $X$ is called infinite-dimensional if it is not contained in a finite-dimensional linear subspace of $X$.
Problem. Let $L$ be an infinite-dimensional subset of the linear ...
- 41.8k
7
votes
1
answer
511
views
Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$
where $\lVert \rVert$ is the ...
- 1,043
7
votes
2
answers
647
views
If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?
Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class.
Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
- 1,054
0
votes
0
answers
96
views
Books on limiting properties of matrices with growing size
This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...
- 101
1
vote
1
answer
195
views
Eigenvalues of operator
In the question here
the author asks for the eigenvalues of an operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
Here I would like to ask if one can extend ...
- 192
1
vote
0
answers
87
views
Computation of the trace of a convolution operator
I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö".
https://iopscience.iop.org/article/10.1070/...
- 197
2
votes
0
answers
81
views
Perturbed Gram matrix
Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix
$$\sum_{t=1}^T(x_t ...
- 215
0
votes
1
answer
357
views
Continuity of point evaluation on space of Hölder functions with $L^p$ norm
Let $L>0$ and $\Omega \subset \mathbb{R}^n$ a bounded Lipschitz domain. Define
$$
B_{\frac12,L}:=\{f \in L^2((0,1) \times \Omega) : \|f(t,\cdot)-f(s,\cdot)\|_{L^2(\Omega)} \leq L|t-s|^{\frac12},~ \...
- 319
1
vote
1
answer
114
views
Is a $1_A \otimes U$ invariant subspace of $\mathcal{H}_A \otimes \mathcal{H}_B$ a product $V_A \otimes \mathcal{H}_B$?
Consider a subspace $V$ of $\mathcal{H}_A \otimes \mathcal{H}_B$, with $\mathcal{H}_A$ and $\mathcal{H}_B$ finite-dimensional Hilbert spaces, that is $1_A \otimes U$ invariant for all unitary ...
- 133
1
vote
1
answer
220
views
Dimension of commutant
Suppose that $A = M_n(\mathbb{C})$ be the algebra of $n*n$ matrices over $\mathbb{C}$.
If com(A) = {$B \in M_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
- 581
4
votes
1
answer
303
views
Which operators on the trace-class operators extend to operators on Hilbert-Schmidt operators?
Let $\mathcal{H}$ be a separable Hilbert space and let $TC( \mathcal{H})$, $HS(\mathcal{H})$ be the space of trace-class operators and Hilbert-Schmidt operators on $\mathcal{H}$. Recall that these ...
- 1,054
4
votes
1
answer
591
views
Derivative of trace
Consider two positive-semi definite matrices $T_1, T_2$ of unit trace.
Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two.
We then study $f(\lambda) := \operatorname{tr}(T(\...
- 536
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
3
votes
3
answers
578
views
When is a linear subspace to be closed in all compatible topologies
Let $V$ be a real vectors space, and $W$ be a linear subspace.
Say $W$ is obviously closed if, for every topology on $V$ that makes $V$ a Hausdorff locally convex topological vector space, the ...
- 537
6
votes
0
answers
107
views
Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 536
7
votes
1
answer
547
views
Is this operator bounded?
Let $T$ be an invertible positive operator and $S$ be another positive operator on a complex Hilbert space.
We then study
$$ \Vert (T+S)^{-1/2}T(T+S)^{-1/2}\Vert$$
I would assume that this norm is ...
- 192
7
votes
1
answer
532
views
Almost commuting matrices, one a projection, is there a nearby projection that commutes?
Suppose that $P, A, Q \in \mathbb{M}^{n \times n}(\mathbb{R})$ (I'm still interested if it must be done over $\mathbb{C}$), (EDIT:) suppose that $A$ is given, $P$ is an orthogonal projection, and $\...
- 393
1
vote
0
answers
61
views
Discrete Orlicz space estimate
We consider the discrete LlogL space of sequences $x=(x_i)$ such that
$$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$
Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL ...
7
votes
1
answer
635
views
Convergence of $\exp(tQ)$ in operator norm as $t\rightarrow\infty$
This is not a homework problem, so I am not sure whether this has a "good" answer or not. I came up with this question when I am now learning functional analysis and wonder whether my "...
- 715