All Questions
Tagged with fa.functional-analysis matrix-analysis
65 questions
3
votes
0
answers
81
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 73
1
vote
1
answer
122
views
distance in the matrix algebra w.r.t. the nuclear norm
Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
- 375
2
votes
1
answer
214
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
1
vote
0
answers
92
views
Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
- 156
0
votes
0
answers
55
views
Johnson-Lindenstrauss type result for matrix factorization
The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
8
votes
2
answers
675
views
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
It is known that an entire function that is nowhere zero must be the exponential of another entire function.
Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
- 286
4
votes
0
answers
120
views
Matrix product of entire functions
Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
2
votes
1
answer
112
views
The eigenvectors of adding a particular rank one matrix to the circulant matrix
Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.
Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
- 4,058
2
votes
1
answer
147
views
Question on density of certain set of matrices
Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I ...
- 286
0
votes
0
answers
124
views
Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with
$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$
How can we compute the eigenvectors of $T$?
- 4,058
0
votes
0
answers
70
views
Follow-up question regarding real singular matrices with additional details
After my question whose answer turned out to be false, I re-examined the course of my proof, which is actually seperate from the one in my question, and found out that there's another condition, at ...
- 286
3
votes
1
answer
521
views
Is the set of real matrices with at least one real logarithm closed under multiplication?
Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
- 286
1
vote
0
answers
106
views
Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 286
0
votes
1
answer
75
views
The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...
- 4,058
3
votes
0
answers
115
views
Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
2
votes
0
answers
43
views
Selecting some linearly independent columns of a particular matrix
Let us consider the matrix $C=A_1+A_2$ where :
$A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$
$A_2$ is the the $n$ by $n$ block ...
- 4,058
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
2
votes
1
answer
226
views
Showing a 2-by-2 matrix is a contraction
Let $S\subseteq\mathbb{T}:=\{z\in\mathbb{C}:\vert z\vert=1\}$ be a compact set such that $\operatorname{conv}S\supseteq\{z\in\mathbb{C}:\vert z\vert\leq\frac{1}{\sqrt{2}}\}$ and $B\in M_2(\mathbb{C})$....
- 231
2
votes
1
answer
179
views
Extension of the projective norm to a cross norm
Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be an operator system. Is it possible to extend the projective norm (the greatest cross norm) ...
- 231
0
votes
1
answer
232
views
Local differentiability of eigenvalues and eigenvectors of a real symmetric matrix
Let $A(x)\in\mathbb{R}^{n\times n}$ be a real symmetric matrix depending on the point $x\in\mathbb{R}^n$, where the eigenvalues are not necessarily simple. Can we say that for all $x$ there exists an ...
- 39
1
vote
0
answers
88
views
2-positivity to 3-positivity
Let $B\in M_3(\mathbb{C})$ and $S_3=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{pmatrix}
$. Suppose $I_3\otimes I_2+B\otimes X+B^*\otimes X^*\geq 0$ for all $2$...
- 231
2
votes
1
answer
151
views
Banach-Mazur distance between Schatten-$p$ classes
Let $M_n$ denote the set of all $n\times n$ complex matrices. Let $1\leq p<\infty.$ For $A\in M_n$ define $\|A\|_p:=(Tr(A^*A)^{p/2})^{1/p}$ where $Tr$ denotes the usual trace of a matrix. Then $\|.\...
- 1,879
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
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
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,$$
where $\lVert \rVert$ is the ...
- 1,043
1
vote
0
answers
70
views
Minimax type principle for a self-adjoint operator acting on a Hilbert space
Let $T\in\mathcal{B}(\mathcal{H})$ be a self-adjoint operator acting on a Hilbert space $\mathcal{H}$. Suppose $k\in\mathbb{N}$. Define $$\lambda_k(T)=\sup\limits_{\substack{\mathcal{M}\subseteq \...
- 231
0
votes
0
answers
90
views
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
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
4
votes
1
answer
208
views
Uniform smoothness inequality for Schatten norms
I've previously asked this question on stack exchange.
I'm looking for a proof of the inequality
$$
\left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^...
- 183
9
votes
2
answers
610
views
When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold?
Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\...
- 93
0
votes
0
answers
49
views
Non-square multiplication operator matrix
Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$.
Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$
Intuitively, $$K: [L^2(0,1)]^n ...
- 617
3
votes
2
answers
370
views
A question of invertibility of matrices
Let $A$ and $B$ be self-adjoint $n \times n$ matrices. Let $A$ be diagonal. Suppose $A+tB$ and $tA+B$ are invertible for all $t \in \mathbb R$. What can we say about $A$ and $B$?
My guess is that $\...
- 1,879
1
vote
1
answer
107
views
A unitary matrix of functions [closed]
If $A(z)=[A_{ij} (z)] $ is an $n\times n$ unitary matrix valued functions. Is there a characterization of such matrix if:
(1) the entries are analytic functions on a set $D$.
and
(2) if the ...
- 39
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
0
votes
0
answers
90
views
Special kind of translation and rotational invariance of the numerical range
Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Is the following statement true?
If $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B\in M_n$ then $W(B\otimes (X+I_n))\subseteq W(B\...
- 231
-1
votes
1
answer
114
views
Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values
To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
- 723
2
votes
2
answers
185
views
Comparison of methods to define a matrix function (Jordan canonical form, Hermite interpolation and Cauchy integral)? [closed]
There are many equivalent ways of defining a function $f(A)$ of a matrix $A$. We focus on Jordan canonical form, Hermite interpolation and Cauchy integral.
What is the difference between methods for ...
- 21
4
votes
1
answer
151
views
Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
- 12.5k
6
votes
0
answers
320
views
Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?
Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
- 596
3
votes
0
answers
86
views
What kind of set is this, spanned by two positive definite matrices?
Let $A$ and $B$ be Hermitian positive definite $n\times n$ matrices over $\mathbb C$ or $\mathbb R$. Then for real $k,\ell,$ the matrix $A^kB^\ell A^k$ is well-defined and again Hermitian positive ...
- 13.4k
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\}. $$
In ...
- 141
3
votes
0
answers
48
views
Questions on "The condition number of a randomly perturbed matrix"
This question is about the two vectors $w'$ and $y$ that are necessary for the argument in section $7$ (page 6) of this paper by Terence Tao and Van Vu,
https://arxiv.org/abs/math/0703307 (that ...
- 2,246
-2
votes
1
answer
158
views
About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246
8
votes
1
answer
522
views
Concavity of the trace of a matrix power
Let $B$ be an $n\times n$ matrix, and define $f$ to be the function that maps positive semidefinite (PSD) $n\times n$ matrices $A$ to real numbers by
$$
f(A) = \mathrm{trace}( (B^*A^2B)^{1/3}).
$$
...
- 757
9
votes
0
answers
261
views
SVD-type decomposition for the tensor product of three Hilbert spaces?
(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the ...
- 25.8k
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) ||...
- 1,465
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} < ...
- 1,465
2
votes
0
answers
125
views
When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?
We have $p \in \mathbb{N}$ and $\mathcal{H}$ is a Hilbert space.
let's consider a matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p} \in \mathcal{S_H}^{p\times p}\!\!\,,$ that is a $p\times p$...
- 195
6
votes
1
answer
1k
views
On proof of the conditionally negative definiteness of a kernel
Let the kernel be $f(\mathbf{x},\mathbf{y}) = \arccos(\mathbf{x}^T \mathbf{y})$, where $\mathbf{x}$ and $\mathbf{y}$ are $\ell_2$ normalized vectors of the same dimensionality, and $\arccos(\cdot): [-...
- 147
21
votes
1
answer
2k
views
Almost commuting unitary matrices
Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
- 901
5
votes
1
answer
136
views
connected stable rank
There is a beautiful formula by Leonid Vaserstein relating the Bass and topological stable rank of a commutative unital Banach algebra A to
that of the matrix algebra M_n(A). Is there something ...
- 687