All Questions
Tagged with linear-algebra operator-theory
74 questions
1
vote
1
answer
151
views
Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 113
4
votes
1
answer
54
views
Krein-Rutman for integral transforms: proof of convergence to leading eigenvector
Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory.
Consider an integral ...
- 312
2
votes
1
answer
215
views
Forming real positive semidefinite matrices from complex matrices
I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices.
Let $Q \in \...
- 31
0
votes
0
answers
64
views
When is a symmetric block Toeplitz matrix invertible?
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
...
2
votes
0
answers
60
views
Perron-Frobenius theory for operators on matrices
Let $A$ be a Hermitian linear operator on the space of $n\times n$ complex matrices. Let's suppose $A$ is "non-negative" in the sense that it preserves the cone of non-negative definite (...
- 53
0
votes
1
answer
142
views
Matrix-order derivatives (differentiating a function a matrix number of times)
I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
1
vote
1
answer
217
views
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 ...
- 4,145
3
votes
1
answer
83
views
Detecting linear operator from actions of powers on subspace
Say I have a sequence of linear operators $A_1,...,A_n$ on a (real) vector space $V_1$. I suspect that there's a second vector space $V_2$, and an operator $A$ on $V_1\oplus V_2$, such that $A_i=(\...
- 4,593
1
vote
0
answers
133
views
Infinite dimensional matrix solvability
In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...
- 573
5
votes
2
answers
276
views
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 ...
- 51
3
votes
0
answers
198
views
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 \...
- 31
1
vote
1
answer
1k
views
Reference for (general case) of uniqueness of singular value decomposition (SVD)
My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values.
I believe that the statements and proofs on this StackExchange posts are ...
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
141
views
On the eigen vectors of a diagonalizable matrix
Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
- 4,058
3
votes
0
answers
77
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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
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
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
2
votes
0
answers
506
views
Finding a basis for the range of a linear function
I realize this question is not high level but I have posted it on Math Stackexchange:
Stackexchange question
and have received some upvotes but no answers or comments, so I am trying here.
I will need ...
- 121
4
votes
1
answer
670
views
Is the product of projection matrices diagonalizable?
Let $P_i=V_{i}V_{i}^{\top}\in\mathbb{R}^{m\times m}$ where $\forall i\in[T]: V_{i}\in\mathbb{R}^{m\times n}$ is a “tall” matrix (i.e., $m \ge n$) with orthonormal columns. Note that these matrices are ...
- 673
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
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
$$
B_{\frac12,L}:=\{f \in L^2((0,1) \times \Omega) : \|f(t,\cdot)-f(s,\cdot)\|_{L^2(\Omega)} \leq L|t-s|^{\frac12},~ \...
- 319
1
vote
1
answer
220
views
Dimension of commutant
Suppose that $A = M_n(\mathbb{C})$ be the algebra of $n*n$ matrices over $\mathbb{C}$.
If com(A) = {$B \in M_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
- 581
8
votes
0
answers
333
views
Monotonicity of log determinant of Gaussian kernel matrix
Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation}
be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ ...
- 55
3
votes
1
answer
415
views
Inverse of block matrix
Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$
Consider the block matrix
$$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$
...
- 192
3
votes
0
answers
234
views
Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $
What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
- 923
4
votes
1
answer
591
views
Derivative of trace
Consider two positive-semi definite matrices $T_1, T_2$ of unit trace.
Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two.
We then study $f(\lambda) := \operatorname{tr}(T(\...
- 536
3
votes
1
answer
740
views
Operator norm of difference of matrix decompositions
This question is in part related to a question that I have already posed.
Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
- 55
7
votes
1
answer
635
views
Convergence of $\exp(tQ)$ in operator norm as $t\rightarrow\infty$
This is not a homework problem, so I am not sure whether this has a "good" answer or not. I came up with this question when I am now learning functional analysis and wonder whether my "...
- 715
0
votes
0
answers
113
views
Error bounds on the expansion of square root of matrix
I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...
- 427
2
votes
1
answer
358
views
The operator equation $AB = \lambda BA$ for self-adjoint operators
Suppose that $A$ and $B$ are self-adjoint bounded linear operators on a Hilbert space and $\lambda \in \mathbb{C}$. It turns out that if $\lambda \notin \{-1, 1\}$ then $AB=\lambda BA \implies AB = ...
- 3,699
11
votes
2
answers
353
views
Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 1,495
4
votes
1
answer
353
views
When is rank-1 perturbation to a positive operator still positive?
Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
- 695
1
vote
0
answers
139
views
Subgradient chain rule
Suppose $$F:\mathbb{R}^n \to \mathbb{R},\; F(x)=\mathrm{max}_\mathrm{eig}(C-\mbox{diag}(x)).$$
I am trying to find a subgradient of $F$ at $x_0$. A subgradient of $\mathrm{max}_\mathrm{eig}$ is given ...
- 11
8
votes
3
answers
691
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
- 12.6k
0
votes
0
answers
87
views
Orthogonal functions and linear operators
Consider the following function $f: [-1,1] \rightarrow \mathbb{R}$, expanded in terms of Legendre functions,
$$
f(y;\boldsymbol{\beta}) = \sum_{i=0}^{\infty} \beta_i P_i(y)
$$
where $\boldsymbol{\beta}...
- 69
9
votes
2
answers
912
views
A Matrix Inequality for positive definite matrices
Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|\leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^{\frac{...
- 1,879
4
votes
2
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159
views
$\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$.
Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality
$$\left\|\displaystyle\...
- 1,154
2
votes
0
answers
257
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 ...
- 620
9
votes
1
answer
535
views
Well known matrix inequality?
I suspect that the following matrix inequality is well known, but I can't find a reference or proof:
Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true?
$${...
- 1,211
2
votes
1
answer
126
views
Determinant of a matrix involving the Prolate Spheroidal Wave Functions
The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation:
$$\int_{-T}^T\varphi_n(x) \text{sinc}(t-x) dx = \lambda_n \varphi_n(t)$$
where $\text{sinc}(t) = \sin(\pi ...
- 93
11
votes
2
answers
2k
views
Operator that commutes with projections
We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$
Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
- 536
1
vote
0
answers
101
views
Operator norm for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$
Suppose $C$ is a $n$ by $n$ real symmetric matrix, and $x\in R^n$. Is there an operator norm of $C$ for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$?
If I decompose $C$ into $A'A = C^{-1}$, It seems ...
- 163
6
votes
1
answer
1k
views
Inequality between nuclear norm and operator norm for positive definite matrices
I will use $\|\|_*$ to denote the nuclear norm (sum of singular values) and $\|\|_2$ to denote the operator norm / matrix 2-norm (largest singular value).
Consider two positive definite $n \times n$ ...
- 61
5
votes
0
answers
330
views
Best Approximation in Operator/non-Frobenius Norm
Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$
$$\...
- 153
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{...
- 43.2k
2
votes
1
answer
1k
views
Norm of a the skew symmetric part of a unitary matrix
Let $U \in \mathbb{R}^{n \times n}$ be a unitary matrix, $U$ can be nonsymmetric, its eigenvalues can be complex numbers and all have modulus $1$.
Is there an upper bound for the maximum singular ...
- 323
4
votes
0
answers
96
views
On decidability of an infinite dimensional dynamic system
Consider two infinite dimensional Toeplitz integer matrices $A,B$ where $B$ is just a shift operator and an infinite dimensional vector $V_0$.
Given a prime $p$ and an infinite dimensional integer ...
- 13.9k
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 ...
- 305
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 $A_n:=I_n-M_n^2$. ...
- 43.2k
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 ...
- 131