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8 votes
1 answer
172 views

Distance between subalgebras and positive elements in matrices

I repost here from stackexchange, as I was not given an answer there. (https://math.stackexchange.com/questions/2956530/distance-between-subalgebras-and-commutants-in-matrix-algebras) This is a ...
Alessandro Vignati's user avatar
4 votes
2 answers
159 views

$\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting

Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$. Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality $$\left\|\displaystyle\...
Student's user avatar
  • 1,154
1 vote
1 answer
1k views

Inequality for the operator norm of a product of matrices

I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that $$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...
BGJ's user avatar
  • 449
2 votes
1 answer
107 views

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 |...
Mark M. Wilde's user avatar
8 votes
2 answers
323 views

Matrix rescaling increases lowest eigenvalue?

Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\...
André's user avatar
  • 225
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 $...
erz's user avatar
  • 5,529
1 vote
1 answer
136 views

Conditions to obtain a real logarithm of a unitary unimodular complex matrix?

The problem statement is the following: $$U=\exp\{iV\}$$ where $U$ is a unitary unimodular matrix of the following form: $$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
john melon's user avatar
8 votes
1 answer
485 views

An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm

Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that $$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\ a_j & b_j & c_j \\ ...
Chen Dan's user avatar
  • 563
26 votes
2 answers
1k views

Symmetric strengthening of the Cauchy-Schwarz inequality

In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle ...
Jochen Glueck's user avatar
51 votes
2 answers
5k views

A strengthening of the Cauchy-Schwarz inequality

Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
Nathaniel Johnston's user avatar
2 votes
0 answers
256 views

The nonlinear operator defined as the commutator of a matrix and a nonlinear operator

In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up: Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...
groupoid's user avatar
  • 620
2 votes
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 ...
user39756's user avatar
  • 141
3 votes
2 answers
137 views

What are some applications of Dilation Structures(idempotent right quasi-groups) from Emergent Algebra?

According to the following Journal Articles, there are these structures called Dilation Structures that are formalised in Emergent Algebras, examined in the case of metric spaces with dilations, and ...
Alexander's user avatar
  • 151
0 votes
1 answer
269 views

Limit of eigenvalues of a matrix perturbation sequence

Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...
Ralph B.'s user avatar
  • 135
11 votes
2 answers
2k views

Operator that commutes with projections

We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$ Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
Sascha's user avatar
  • 536
-1 votes
1 answer
132 views

About a property in a reflexive Banach space

Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) ...
MSMalekan's user avatar
  • 2,118
13 votes
2 answers
653 views

The geometry of $\mathbb{R}^n$

Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces. Then we define the set of equivalence classes $$G(X,Y):=\left\{[T]; T,S \in ...
Sascha's user avatar
  • 536
9 votes
2 answers
338 views

Does $End(V)$ remember $V$, where $V$ is a locally convex space?

Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a ...
André Henriques's user avatar
5 votes
1 answer
607 views

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^\...
naughie's user avatar
  • 177
3 votes
0 answers
422 views

Isometries between subspaces of finite-dimensional vector spaces

I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this. Taking $n \le m$, one ...
Dave's user avatar
  • 31
35 votes
2 answers
2k views

Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?

Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
David E Speyer's user avatar
12 votes
1 answer
229 views

History of publication of von Neumann's characterization of orthogonally invariant matrix norms

Von Neumann has a result (rather well-known in convex analysis circles) which states that every orthogonally invariant matrix norm (meaning $\| P M Q\| = \| M \|,$ for any orthogonal $P, Q$) is a ...
Igor Rivin's user avatar
  • 96.4k
4 votes
0 answers
141 views

algebraic connectivity of a tree

Suppose that $T$ is a tree with $n$ vertices and $L$ is the Laplacian matrix of $T$ and $0=\mu_1 \leq \mu_2 \leq \cdots \leq \mu_n$ are laplacian eigenvalues. I think the multiplicity of $\mu_2$ can ...
MH.Fakharan's user avatar
1 vote
1 answer
291 views

Norm of solution of quadratic program

In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}_{m\times n}$ and $q\in\mathbb{R}^n$. Define $$x^* ...
Conner DiPaolo's user avatar
1 vote
2 answers
400 views

Reference request for Stieltjes Transform

I am wondering to use Stieltjes transform to signal processing like Fourier Transform. The Fourier is known to give the frequencies of a signal but not sure what Stieltjes transform gives. I am only ...
Mainag's user avatar
  • 27
4 votes
1 answer
192 views

When does iteration of an infinite Toeplitz matrix converge?

Consider a Toeplitz matrix $T$, indexed by $\mathbb{N}_0 \times \mathbb{N}_0$. given by the sequence $t_k,k \in \mathbb{Z}$ where $t_k \geq 0,\sum_{k=-\infty}^\infty t_k=1$. By this I mean that $T_{i,...
Ian's user avatar
  • 325
3 votes
1 answer
336 views

The generalization of commutative property of orthogonal projectors on a subspace to the whole space

Let for $i\in [n]$, $P_i$ be some orthogonal projectors defined on the Hilbert space $W$ such that they commute on subspace $V < W$ (i.e, for any $i, j \in [n]$ and $v \in V$: $P_iP_j(v) = P_jP_i(v)...
Guest's user avatar
  • 31
5 votes
1 answer
1k views

The spectrum of the discrete Laplacian

Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$ On this we ...
Dr. House's user avatar
0 votes
1 answer
110 views

Number theory for operator bound

Let $\gamma_i$ be such that for even $i$ $\gamma_i=1$ and for odd $i$ $\gamma_i$ shall have absolute value $1$ and the product of all of the odd ones is also on the complex unit circle but not 1 or -1....
Zinkin's user avatar
  • 501
1 vote
2 answers
693 views

Separating convex sets in Vector spaces

This question just popped on my mind. Let $A, B$ two disjoint, nonempty convex sets in the vector space $X$, can they be separated via a nonzero linear function in $X' = \{ f : X \to R ~ | \quad \...
Red shoes's user avatar
  • 369
3 votes
2 answers
471 views

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 ...
Dima Pasechnik's user avatar
2 votes
1 answer
226 views

Tighest Gap $\|x\|_1/\|x\|_2$ between $\ell^1$ and $\ell^2$ norms

I'm looking specifically at the optimization problem $$ \begin{align*} \text{maximize: }& n - \frac{\|\lambda\|_1^2}{\|\lambda\|_2^2}\\ \text{subj. to: }& \lambda \succeq \epsilon\mathbf{1} \...
Conner DiPaolo's user avatar
5 votes
0 answers
330 views

Best Approximation in Operator/non-Frobenius Norm

Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$ $$\...
Conner DiPaolo's user avatar
5 votes
4 answers
839 views

Norm bounds on spectral variation and eigenvalue variation

Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively. The spectral variation of $B$ w.r.t. $A$ and the eigenvalue variation of $B$ and $A$ are, respectively, \begin{...
T. Amdeberhan's user avatar
3 votes
1 answer
154 views

Solving Matrix/Operator Equation $H P X + X P H + HQ = 0$

This problem arises when minimizing the operator equation $X P X^* + X Q + R$ over positive $X$ with respect to the positive cone on a Hilbert space $\mathcal{H}$. The (reduced) task: Given $P$ and $...
Conner DiPaolo's user avatar
8 votes
1 answer
353 views

$E_n(\ell^\infty)=SL_n(\ell^\infty)$?

Let $R$ be a commutative unital ring $R$ with unit element $1$. For $n\in \mathbb{N}=\{1,2,3,\cdots\}$, let $SL_n(R)$ be the group of all $n\times n$ matrices with entries from $R$ having ...
KevinC's user avatar
  • 81
4 votes
1 answer
381 views

Continuous linear combination of continuously varying vectors?

Let ${\bf{e}}_1, {\bf{e}}_2, {\bf{e}}_3:[0,1]\rightarrow \mathbb{R}^3$ be continuous, $\mathbf{0}\neq \mathbf{v}\in \mathbb{R}^3$. Suppose that the following condition (C) holds: $$ \exists d>0: ...
Magnus's user avatar
  • 81
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 ...
Kinzlin's user avatar
  • 305
8 votes
1 answer
386 views

Lower bound for $\frac{\sum_{i,j}\min((f_i-f_j)^2,(g_i-g_j)^2)}{\sum_{i,j}\max((f_i-f_j)^2,(g_i-g_j)^2)}$

Let $f\in\mathbb{R}^n$ and $g\in\mathbb{R}^n$ be two orthogonal unit vectors such that $\sum_{i}{f_i}=\sum_{i}{g_i}=0$. Question. Can we prove this? $$\frac{\sum_{\{i,j\}}\min((f_i-f_j)^2,(g_i-...
j.s.'s user avatar
  • 519
11 votes
2 answers
714 views

A neat evaluation of an infinite matrix?

Let $M_n$ be an $n\times n$ matrix defined as $$M_n =\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$ With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
T. Amdeberhan's user avatar
8 votes
1 answer
576 views

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 ...
LCO's user avatar
  • 506
4 votes
2 answers
671 views

When does this linear matrix equation have a unique symmetric, positive definite solution?

I encountered the following matrix equation for $A, N, Q \in \mathbb{R}^{n \times n}$ and $A^T=-A$ and $N^T=-N$ $$[X,A]+N^TXN+Q = 0$$ where $Q$ is symmetric, positive definite. My final goal is to ...
Joppi's user avatar
  • 41
3 votes
0 answers
588 views

Norm in a product vector space induced by a norm in $\mathbb{R}^d$

I posted this question originally here (nobody answered there): https://math.stackexchange.com/questions/2066318/is-the-following-function-a-norm Let $\| \|$ be any norm in $\mathbb{R}^d$. Consider ...
William M.'s user avatar
2 votes
0 answers
92 views

Estimating the size of a subset of $\mathbb{R}^N$

This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
Kevin Smith's user avatar
  • 2,480
3 votes
0 answers
119 views

Increasing sequence of closed subspaces of $L^2$ and error estimate of a product of orthogonal projections

We define an increasing sequence of closed subspaces \begin{align*} V_{0} \subset V_{1} \subset V_{\ell} \subset \dots \end{align*} of $L^2(I)$ where $I=(0,x_{max})$, and each $V_{\ell}$ is equipped ...
user144209's user avatar
4 votes
3 answers
3k views

Is this inequality involving the Frobenius norm right?

Let $A$ be a generic (or varying) square, real $ n \times n$ matrix. Let $G$ be a fixed $n \times k$ matrix, $k < n.$ Denote by $||.||_F$ the Frobenius norm. Is it true that $||AG||_F \geq c(G) ||...
Learning math's user avatar
2 votes
0 answers
147 views

Is the following inequality true for the norm of Moore-Penrose pseudoinverses?

Let $L$ be a real, positive semi-definite, symmetric, square matrix, with pseudoinverse $L^{+}$. It can be shown for the operator norms $||.||_{op}$ that: if $L$ is invertible and $||I - L||_{op} < ...
Learning math's user avatar
1 vote
1 answer
158 views

When do we have $B_Y\subset T(B_X)$ if and only if $\overline{B_Y}\subset T(\overline{B_X})$?

Let $X$,$Y$ be normed spaces, $T:X\to Y$ be a bounded linear operator. Denote the open and closed unit balls by $$ B_X:=\{ x\in X\ |\ \|x\|<1\} \\ \overline{B_X}:=\{ x\in X\ |\ \|x\|\le1\} $$ and ...
BigbearZzz's user avatar
  • 1,245
0 votes
1 answer
198 views

The eigenfunctions of an operator commuting with all rotations.

When reading the paper E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374 I found an argument like the following. Given an bounded and self-adjoint linear operator ...
gregarki khayal's user avatar
3 votes
0 answers
261 views

Generalization of trace and associated determinant

The standard relation between the trace and the determinant of matrices is presented in the MO-Q "Cycling through the zeta garden" where the log and exp functions allow one to jump between additive ...
Tom Copeland's user avatar
  • 10.5k

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