All Questions
Tagged with fa.functional-analysis linear-algebra
325 questions
0
votes
0
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90
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Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
2
votes
1
answer
133
views
Control on dimension of image
Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
- 5,405
2
votes
0
answers
66
views
Minimizing a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
5
votes
1
answer
332
views
On a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
1
vote
0
answers
70
views
Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix
Let $\phi:[-1,1] \to \mathbb R$ be a function such that
$\phi$ is $\mathcal C^\infty$ on $(-1,1)$.
$\phi$ is continuous at $\pm 1$.
For concreteness, and if it helps, In my specific problem I have $\...
- 6,853
2
votes
0
answers
99
views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
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
1
vote
1
answer
100
views
$\ell^1$-bound on graph laplacian with weight
Consider the $\mathbb Z^2$ lattice, we then define for $u=(u_{ij})_{i,j \in \mathbb Z}$ the discrete Laplacian
$$(\Delta u)_{i,j}=u_{i+1,j}+u_{i-1,j}+ u_{i,j+1}+u_{i,j-1}$$
and the weight which pushes ...
- 536
0
votes
0
answers
113
views
Error bounds on the expansion of square root of matrix
I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...
- 427
16
votes
1
answer
537
views
Balls in Hilbert space
I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
- 2,049
0
votes
0
answers
45
views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,405
2
votes
0
answers
114
views
Is $A$ is small on bounded functions, is there a large subdomain on which $A$ is small?
Let $A$ be a symmetric linear operator of norm $\leq 1$ on the space of functions $f:S\to \mathbb{R}$, where $S$ is a set with $N$ elements. Define the inner product $\langle \cdot,\cdot\rangle$ by ...
- 20.2k
7
votes
0
answers
107
views
Potential p-norm on tuples of positive operators
This is a follow-up to this question on p-norms of tuples of operators.
Consider $\left[\begin{matrix} A \\ B \end{matrix}\right] \in B(H)^2_+$, meaning $A,B\geq 0$, and define
$$
\left\|\left[\begin{...
- 3,984
6
votes
1
answer
236
views
Potential p-norm on tuples of operators
Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define
$$
\left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|_p = \| |A|^p + |B|^p\|^{1/p}.
$$
Q: Is this a norm?
...
- 3,984
6
votes
1
answer
299
views
Continuity of eigenvectors
Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \...
- 536
5
votes
2
answers
1k
views
Compact operator without eigenvalues?
Consider the operator $M$ on $\ell^2(\mathbb{Z})$ defined by for $u\in \ell^2(\mathbb Z)$
$$Mu(n)=\frac{1}{\vert n \vert+1}u(n).$$ This is a compact operator!
Then, let $l$ be the left-shift and $r$ ...
- 173
-1
votes
1
answer
215
views
Dense linear span implies closed convex hull has non-empty interior
Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
- 5,405
3
votes
1
answer
115
views
Approximation of vectors using self-adjoint operators
Let $T$ be an unbounded self-adjoint operator.
Does there exist, for any $\varphi$ normalized in the Hilbert space, a constant $k(\varphi)>0$ and a sequence of normalized $(\varphi_n)$ such that $$...
- 173
4
votes
0
answers
109
views
Characterization of "PSD-Squared" Matrices
$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...
- 6,351
-1
votes
1
answer
323
views
Expressing the sum of two squared inner products more compactly: is it possible to lift the dimension? [closed]
Let $v_1,v_2\in\mathbb{R}^d$ be two fixed vectors, and $\langle \cdot,\cdot\rangle_{\mathbb{R}^d}$ be the usual Euclidean inner product in $\mathbb{R}^d$.
My question is as follows. Is there an (...
- 1,096
2
votes
0
answers
58
views
Is keeping the kernel fixed an open condition for maps of vector bundles?
More precisely, let $M$ be a smooth manifold, $E_1$, $E_2$ vector bundles over $M$, and consider a $C^\infty(M)$-linear map $A:\Gamma(E_1) \to \Gamma(E_2)$ of vector bundles.
Now consider the ...
- 73
2
votes
1
answer
153
views
Regarding upper semicontinuity of a function
Let $E$ be a linear subspace of $\mathbb{C}^{n\times n}$.
Define the function $\mu_E:\mathbb{C}^{n\times n}\longrightarrow \mathbb{R}_+$ as
$$
\mu_E(A)=\frac{1}{\inf\{\|X\|:X\in E\text{ and }\det(I_n-...
- 31
0
votes
0
answers
99
views
Link between eigenvalues of a symmetric matrix and a functional space
Let $f_1,\dots,f_n \in L^2(\mathbb{R},\mathbb{R})$ be $n$ mutually orthogonal functions with $\int f^2_i =1$ such that $|\{x \in \mathbb{R} | f_i(x) = 0\}| = 0$ for any $i \in \{1, \dots,n\}$. Does ...
- 31
1
vote
0
answers
448
views
Smallest eigenvalue for large kernel matrix
I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$.
...
- 151
9
votes
1
answer
511
views
Do these surfaces intersect?
For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$
with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$,
does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
- 91
0
votes
1
answer
126
views
Tauberian operators
Let $X$ be a Banach space non reflexive and $T$ from $l_2(X)$ to $l_2(X)$ a bounded operator defined by:
$$T(x_n )=\frac{x_n }{n}.$$
We know that :
$$T^{**-1}(l_2(X))=\{x_n^{**} \in l_2(X^{**}) : \...
0
votes
0
answers
255
views
Span of a nonlinear function
Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, ...
- 159
3
votes
2
answers
307
views
Random matrix is positive
This is a follow up question on my previous question here that was on solved in the deterministic setting by Denis Serre, when the perturbation can be separated. Therefore, I decided to split the ...
- 536
4
votes
0
answers
144
views
A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
- 128k
2
votes
3
answers
216
views
Equivalence of operators
let $T$ and $S$ be positive definite (thus self-adjoint) operators on a Hilbert space.
I am wondering whether we have equivalence of operators
$$ c(T+S) \le \sqrt{T^2+S^2} \le C(T+S)$$
for some ...
- 23
8
votes
3
answers
691
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
- 12.5k
4
votes
1
answer
174
views
A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 5,529
3
votes
0
answers
95
views
Sparse perturbation
Let $x, x_0\in\mathbb{R}^n$ be two vectors satisfying $$\frac{\|x\|_1}{\|x\|_2}\leq\frac{\|x_0\|_1}{\|x_0\|_2}.$$
$\| \cdot\|_1$ and $\| \cdot\|_2$ are the $\ell_1$ and $\ell_2$ norm in $\mathbb{R}^n$,...
- 31
3
votes
1
answer
456
views
Duality of Topological Vector Spaces
Let $K$ be a topological field. Let $\text{top-} K \text{-vect}$ be the category of topological $K$-vector spaces $V$, so that the maps $\cdot : K \times V \rightarrow V$ and $+ : V \times V \...
user30211
8
votes
1
answer
678
views
Inequality involving tensor product of orthonormal unit vectors
Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and ...
- 1,495
4
votes
2
answers
871
views
Decay of eigenfunctions for Laplacian
Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$.
Its eigendecomposition is fully known:
see wikipedia
It seems like the largest eigenvalue $\lambda_1$ is ...
2
votes
0
answers
136
views
Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
6
votes
1
answer
203
views
How to calculate the volume of a section of a convex body?
The following is essentially a partial case for my previous question.
Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$....
- 5,529
5
votes
0
answers
350
views
How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
1
vote
0
answers
48
views
Is there an easy characterisation (perhaps some generalised Löwner representation) for operator monotone functions of order $n$?
As per my understanding, roughly stated, $f$ is an operator monotone function of order $n$ if for all $n\times n$ (Hermitian) matrices, $X,Y\ge0$ which satisfy $X\ge Y$, we have $f(X)\ge f(Y)$.
If $f$...
- 131
2
votes
0
answers
240
views
Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
6
votes
2
answers
251
views
uniform approximation by a particular set of functions
Consider the interval $[0,1]$ and let $\mu_k(t)$ with $k=1,\ldots,n$ be continuous functions such that they are all strictly increasing on the interval $[0,1]$ and such that $\mu_1(t)<\mu_2(t)<\...
- 4,135
9
votes
1
answer
953
views
Is there always a complete, orthogonal set of unitary matrices?
The set of size-$n$ unitary matrices span $\Bbb C^{n \times n}$ (this can be proven nicely using polar decomposition). If we select a maximal linear subset of unitary matrices, then we have a basis ...
- 356
2
votes
0
answers
57
views
The significant role of dual frames in the progress of Frame theory
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
- 21
3
votes
0
answers
163
views
Perturbation theory compact operator
Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...
user136577
7
votes
1
answer
743
views
Is Gram-Schmidt on a separable Hilbert space operator norm continuous?
Let $\mathcal H$ be a separable Hilbert space, with inner product $\langle\cdot,\cdot\rangle$, and with orthonormal basis $(e_i)_{i\in\mathbb N}$. Consider a continuous linear embedding $A\colon\...
- 320
1
vote
0
answers
54
views
When does the multi-spectral radius coincide with the spectral radius of the sum of linear transformations?
Suppose that $X$ is a finite dimensional Hilbert space and
$A_{1},\dots,A_{r}:X\rightarrow X$ are linear transformations. Define the multi-spectral radius $\rho(A_{1},\dots,A_{r})$ to be
$$\limsup_{n\...
3
votes
1
answer
791
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
5
votes
1
answer
218
views
Spectral radius for multiple linear operators
Suppose that $X$ is a finite dimensional Hilbert space. Let $A_{1},\dots,A_{r}:X\rightarrow X$ be linear operators. Then define the multi-spectral radius of $(A_{1},\dots,A_{r})$ to be
$$\limsup_{n\...
3
votes
1
answer
2k
views
When is the matrix norm multiplicative
Let $|| = ||_{p,q}$ be an operator norm on $\mathbb R^{n \times m}$. In General, $\|AB\|\le \|A\|\|B\|$. Is there some criterion on $A, B$ (at least for some operator norms) so that $\|AB\| = \lVert ...
- 133