All Questions
Tagged with hilbert-spaces linear-algebra
49 questions
0
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0
answers
46
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What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
- 342
2
votes
1
answer
226
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Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?
I am trying to prove that the function:
$$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$
is a positive definite function over the natural numbers. What has sometimes ...
- 23.7k
1
vote
1
answer
138
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Orthogonal vectors translation using standard vectors
When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$
$$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$
$$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$
It is ...
1
vote
1
answer
217
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Perturbation of matrices
Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$.
Question. Does there exist a Lebesgue measurable ...
- 1,869
3
votes
0
answers
136
views
Unitary operators with the same inner product as vectors
Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...
- 193
2
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0
answers
258
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Orthogonal complement of arbitrary intersection of Hilbert subspaces
Let $H$ a Hilbert space, and $\mathcal C$ an arbitrary set of closed subspaces of $H$. Is it true that
$$\left( \bigcap_{Z\in \mathcal C}Z\right)^\perp = \overline{\sum_{Z\in \mathcal C} Z^\perp}$$
...
- 63
3
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1
answer
67
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Why does the normalization term disappear when computing the MLE of decomposed Gaussians
Computing the Maximum Likelihood Estimator of Gaussians in arbitrary finite Hilbert spaces seems no easy task and I must admit to lamentably fail at it. The classical theory most often relies on ...
- 128k
6
votes
1
answer
300
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Log-convexity of determinant
Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
CommunityBot
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4
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0
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98
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A question on products of linear combinations of complex matrices
Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds
$$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{...
- 24.2k
5
votes
2
answers
276
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Dilation of bounded linear operators
Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
- 3,984
3
votes
0
answers
198
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On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 128k
2
votes
0
answers
134
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How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]
I noticed that in the paper [1] this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...
- 205
1
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0
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94
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Invariance signature in infinite dimension
Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that:
$g_0$ is positive-definite
$g_t$ is non-degenerate for ...
- 395
8
votes
1
answer
446
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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 ...
- 1,453
7
votes
4
answers
260
views
Existence of a non-null and non-negative vector in $F\cup F^\perp$
In $\mathbb{R}^n$ ($n\ge 1$) endowed with the usual dot product, for any linear subspace $F$, does there exist a non-null vector with non-negative coordinates in $F\cup F^\perp$?
- 259
1
vote
1
answer
247
views
Linearity of the directional derivative of a convex functional at the minimum
Let $H$ be a Hilbert space, $T_+(H)$ the set of positive self-adjoint trace-class operators on $H$, and $f : T_+(H) \to [0,m]$ a non-negative, bounded, convex functional. I don't necessarily know that ...
- 6,367
1
vote
1
answer
90
views
A linear algebra question for semi-Euclidean norm
Let us consider Minkowski inner product on $\mathbb R^{1+n}$, defined by
$$ \langle v,w \rangle = -v_0w_0+\sum_{j=1}^n v_j w_j\quad \,\forall\, v,w \in \mathbb R^{1+n}.$$
We say that a vector $v$ is ...
- 39k
5
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0
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471
views
A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?
In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
- 597
-1
votes
1
answer
323
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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
1
vote
0
answers
67
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Spectral theorems for generalized Hermitian matrices
Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...
- 4,547
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 ...
- 61.9k
0
votes
0
answers
87
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Orthogonal functions and linear operators
Consider the following function $f: [-1,1] \rightarrow \mathbb{R}$, expanded in terms of Legendre functions,
$$
f(y;\boldsymbol{\beta}) = \sum_{i=0}^{\infty} \beta_i P_i(y)
$$
where $\boldsymbol{\beta}...
- 69
2
votes
1
answer
107
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tensor stability of block-positive matrices
Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation)
$\langle \psi |...
- 18.7k
2
votes
1
answer
267
views
volume of parallelotope in $L^2(\mathbb R).$ [closed]
Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product.
Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g.,
$$\{ f(...
- 28k
5
votes
1
answer
379
views
Hilbert representation of a bilinear form
Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...
- 7,893
2
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0
answers
246
views
Decay rate of least eigenvalue of Gram matrices
Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$:
$$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$
In ...
- 141
5
votes
1
answer
607
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Does eigenvalue exist in a Hilbert space? [closed]
In a lecture on Quantum mechanics, the professor concluded that if $a$ is a linear operator with $[a, a^\dagger] = 1$, where $a^\dagger$ is the adjoint of $a$ and $[a, a^\dagger] = aa^\dagger - a^\...
- 23k
42
votes
4
answers
33k
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What is the intuition for the trace norm (nuclear norm)?
I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense ...
- 415
3
votes
2
answers
471
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inner product on matrix spaces of multivariate polynomials?
Let $H_{n,d}=\mathbb{R}_d[x_1,..,x_n]$ be the space of $n$-variate homogeneous degree $d$ polynomials, $D=D^\top\in \mathbb{N}^{m\times m}$ a symmetric $m\times m$ matrix. Consider the space $P_D$ of ...
- 8,597
2
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0
answers
126
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Left multiplication Fréchet Differentiable
Let $L^2$ be a separable Hilbert space and let $\{e_i(t)\}_{i \in \mathbb{N}}$ be a basis for it. Moreover let $\phi(T,z)$ and $\psi(T,z)$ be 1-parameter families (in z) in $L^2$, with basis ...
CommunityBot
- 1
0
votes
0
answers
263
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 63.9k
8
votes
1
answer
576
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On the definition of Hilbert spaces and real structures on Hilbert spaces
Let us consider the space $L^2:=L^2(\mathbb{R}^n,\mathbb{C})$ and the associated scalar product $S(f,g):=\int f \overline g$. In distribution theory, we have a situation where we have to deal with two ...
- 42.8k
1
vote
0
answers
94
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Space spanned by pointwise squares of basis functions
Consider the Hilbert space $L^2(\Omega)$ over some Euclidean domain $\Omega$.
Let $F=\{f_i;i\in\mathbb N\}$ be an orthonormal basis of this space consisting of functions in $L^2(\Omega)\cap L^4(\Omega)...
- 8,086
1
vote
0
answers
76
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Linear independence of an odd set of measurable functions
Let $g(t)$ be a convex positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}...
- 5,405
0
votes
1
answer
307
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Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators
Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...
- 195
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0
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252
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Hadamard product (Schur product) in $L^2[0,1]$
Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
- 195
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1
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87
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Finding a vector representation for a data where we only know the inner products
I am an engineer working on speech signal processing and I have a problem that I have encountered while trying to model speech signals. The mathematical formulation is not entirely pure and I try to ...
- 8,491
7
votes
2
answers
7k
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Dual operators between Hilbert spaces: with or without Riesz representation
Let $X$ and $Y$ be Hilbert spaces over the real numbers (so complex conjugation plays no role, and everything will be linear in the strict sense). Let $f : X \rightarrow Y$ be a linear continuous ...
- 105k
1
vote
3
answers
684
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Norm of an operator formed using a unitary operator
Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
- 817
1
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0
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517
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Complex conjugate and unitary complex conjugate
Definition: Let V be complex finite dimensional inner product space
Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$
Definition: Anti-unitary Complex ...
CommunityBot
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3
votes
1
answer
275
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Sum of two parts of a continuous stochastic process
Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all $...
CommunityBot
- 1
5
votes
1
answer
372
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The Maximal $\ell_2$ norm of a signed sum of vectors
Suppose we have $n$ vectors in $\mathbb{R}^n.$ Consider the signed sum of these vectors:
$$U(s_1,\ldots,s_n)=s_1 v_1+s_2 v_2 + \ldots + s_n v_n$$
where $s_j$'s can only take values of $+1$ or $-1.$ I ...
- 131
4
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0
answers
404
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Hilbert Schmidt Operators and the Conditional Expectation Operator
Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
- 289
6
votes
2
answers
744
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Unicity of a vector space frame's dual frame
The Wikipedia page on vector space frames gives a construction to find a dual frame for a given frame. Specifically, given a set of vectors $\{ e_k \}$ in a Hilbert space $\mathcal{H}$ such that for ...
- 660
2
votes
1
answer
1k
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Coercive Symmetric Bilinear form on a Hilbert space
I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance.
1) Consider a continuous symmetric ...
- 882
2
votes
1
answer
386
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Decomposing bilinear forms in Hilbert spaces
You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
- 42.8k
2
votes
1
answer
187
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Partial isometries making families of linearly independent vectors orthogonal
Suppose I have a family of $n$ linearly-independent elements $v_i$ of the Hilbert space $\mathbb{C}^m$, which are not necessarily orthogonal. Can I always find a partial isometry $f: \mathbb{C} ^m \to ...
- 9,486
2
votes
1
answer
205
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Do unitary bijections act invariantly on irreducible representations?
Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{...
- 25.5k
1
vote
3
answers
206
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Linear space of translatable functions.
What are the functions $f$ so that a set $\{a \cdot f(x+b) : a \in \mathbb{R}, b \in \mathbb{R}\}$ is a finite dimensional linear vector space ?
Is there a complete characterization of such functions?...
- 14.5k