All Questions
866 questions
3
votes
0
answers
82
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
0
answers
111
views
References on the partial trace
For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows :
$$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
- 1,399
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
1
vote
1
answer
112
views
Bounding a Riemann sum by its integral limit?
Let $M_{n}(\mathbb{C})$ denote the space of complex $n \times n$ matrices and, for $a>0$, $a \in \mathbb{R}$ fixed, let $A: [0,a) \to M_{n}(\mathbb{C})$ be a given function. I will write $A(t) = (...
0
votes
0
answers
50
views
Eigenvalues of functions on finite discrete sets
Suppose I have an arbitrary function on a finite and discrete set $S$ defined as
$$f: S \times S \to \mathbb{C}^{|S|\times |S|}.$$
The $|S| \times |S|$ matrix $M$ is defined as
$$(M)_{ij}=f(s_i, s_j) \...
- 1
1
vote
0
answers
93
views
Similarity of non-standard matrices
I am researching numerical methods for PDEs. I particular, I am looking at methods for the linear hyperbolic PDE
$$
u_t+au_x=0.
$$
This is a common approach, because successful methods for this model ...
- 111
2
votes
0
answers
214
views
A conjectured generalization of Marcus's inequality
Note: I have edited the post below in order to include sharper (conjectured) inequalities, using $|H_1 \cap H_2|$.
Let $[n] = \{1, \dots, n\}$ and let $\sim$ be an equivalence relation on $[n]$. Then $...
- 5,215
1
vote
2
answers
86
views
Entrywise $\infty$-norm of squared difference of square roots of matrices
For a positive $n \times n$ definite real matrix $M$ we denote by $\sqrt{M}$ the positive square root of $M$. For an $n \times n$ matrix $A$ denote its entrywise infinity norm by
$$\|A\|_{\infty,\...
- 177
1
vote
1
answer
122
views
distance in the matrix algebra w.r.t. the nuclear norm
Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
- 375
0
votes
1
answer
127
views
Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
- 27
0
votes
0
answers
68
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
1
vote
1
answer
151
views
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$
Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
- 27
4
votes
1
answer
199
views
First derivative of $f(A) = \frac{1}{\lambda_{\min}(A)}$ for perturbed matrix
I am working with the matrix function
$$
f(A) = \frac{1}{\lambda_{\min}(A)},
$$ where $A \in \mathbb{R}^{n \times n}$ is a positive definite matrix and $\lambda_{\min}(A)$ is its smallest eigenvalue. ...
- 91
2
votes
0
answers
68
views
Nonlinear random matrix equations
Let $C$ be a matrix; $v$ be a column vector;
$P$, $\Delta$ are random matrices;
$x$ is a random column vector.
$$Cvv^T - \mathbb{E}[P^Tx]v^T - \mathbb{E}[P^Tx v^T \Delta] + O(\Delta^2)= 0$$
$$C^TCv - ...
- 21
3
votes
0
answers
109
views
Faithful traces on reduced $C^*$-algebra of a measured groupoid
Let $G$ be a measured étale groupoid with quasi-invariant measure $\mu$ (that induces the reduced $C^* $-algebra, meaning it has full support) with associated equivalent measures $\nu,\nu^{-1}$.
Is ...
- 261
7
votes
1
answer
118
views
Bound on when sequence of norms of matrix powers starts to decrease
If a matrix $A$ has spectral radius $\rho(A)<1$, it is well-known that $A^n\to0$ as $n\to\infty$, or equivalently $\lVert A^n\rVert\to 0$ for some matrix norm $\lVert\cdot\rVert$; however, it may ...
1
vote
1
answer
153
views
How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
5
votes
1
answer
175
views
Orthogonal projection onto cones in inner product spaces
Let $H_n$ denote the space of $n\times n$ Hermitian matrices. For every $A\in H_n$, using the spectral decompostion of $A$,
$$A=\sum_i \lambda_i x_ix_i^*,$$
one can define the positive and negative ...
- 4,484
10
votes
3
answers
455
views
When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 145
10
votes
0
answers
225
views
Can the trace be computed in any Schauder basis?
I'm cross-posting this question from Math.SE, as it didn't get much attention there.
Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
- 233
3
votes
0
answers
98
views
Bounding the norm of a sum of fourth-order Gaussian vectors
Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$, where $\Sigma = \operatorname{diag}(\lambda_1, \cdots, \lambda_k)$ and
$\lambda_1 \ge \cdots \ge \...
- 97
1
vote
1
answer
111
views
Matrix concentration inequality for unbounded (sub-exponential) matrices
Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$. We aim to establish a high probability bound on the eigenvalues $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ ...
- 97
2
votes
1
answer
237
views
Geometric interpretation of trace of a linear operator
This question is really an addendum to Geometric interpretation of trace
There is a nice account of the trace in Chris Doran's thesis here: http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/...
- 21
7
votes
1
answer
271
views
Existence of a linear map resulting in the determinant being an elementary symmetric polynomial
Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
- 6,351
2
votes
0
answers
160
views
An "almost" true inequality for Hermitian matrices
Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality:
$$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
- 593
7
votes
1
answer
390
views
Questions on symmetric Hadamard matrices
Definitions: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal.
If $A$ is a symmetric matrix, then $A = A^T$ and if $...
- 696
1
vote
0
answers
28
views
explicity quantum expander
A set of $d\times d$ unitary matrices $\{U_1,\cdots,U_n\}$ is called a quantum expander if the identity matrice is the only eigenvector of the linear map $\mathcal{M}(X)=\frac{1}{n} U_i X U_I^{+}$ ...
- 1,503
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
3
votes
1
answer
286
views
Unitary transformations of Vandermonde matrices forms a smooth manifold?
The space of all Vandermonde matrices $V$ with $r$ variables and degree $n$ (as below) forms an embedded submanifold of $\mathbb{R}^{(n+1) \times r}$ when $x_{i} \in \mathbb{R}$. It is naturally a ...
- 275
1
vote
1
answer
87
views
Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?
The spectral theorem says that for every Hermitian matrix $A \in \mathbb{C}^{n \times n}$ there is a unitary matrix $V \in U(n)$ and a diagonal matrix $D \in \mathbb{R}^{n \times n}$ such that $A = ...
- 1,295
5
votes
1
answer
196
views
What is the "natural" or "physical" norm on the Hessian matrix (and other higher derivatives)?
Let $u : \mathbb R^n \rightarrow \mathbb R$ and let $H : \mathbb R^n \rightarrow \mathbb R^{n \times n}$ be its Hessian matrix. What is the "natural" choice of pointwise norm on the Hessian ...
- 641
2
votes
0
answers
85
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
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\...
0
votes
0
answers
43
views
Given two rectangular matrices and they yield the same results when they are multiplied by their own transposes. What can we say about them?
Suppose we have $MM^T = NN^T$, where $M$ and $N$ are both $n$ by $d$ matrices. Assume that $n$ is (much) larger than $d$, are there anything we could conclude about $M$ and $N$, aside from that $N$ ...
- 9
1
vote
0
answers
29
views
Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?
Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
- 11
1
vote
0
answers
21
views
Correlation Matrix Problem of Three Decomposition Level of DWT
I'm trying to apply a DWT with 3 composition levels and the following question arose when calculating the composition matrix.
The step I'm trying to follow is:
The DWT coefficientes are obtained from ...
- 11
2
votes
1
answer
87
views
Symmetric and anti-symmetric matrices and maximal eigenvalues
Suppose we start with a symmetric $n \times n$ matrix $A$, the elements of which are either $1$ or $0$. All the diagonal elements of this matrix are set to be $0$. So, $\lambda_{\text{max}}=\sup \...
- 1,755
0
votes
1
answer
91
views
Matrix quantization and effect on singular values
Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for
$$
\|
\sigma_i(A)-\...
- 5,405
0
votes
0
answers
35
views
What is the impact of individual estimate on each other in matrix inversion?
I am looking to understand the impact of each estimate on each other in matrix inversion.
Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...
- 1
1
vote
0
answers
74
views
Trace map for universal bundle of Grassmannian
Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on ...
- 129
10
votes
3
answers
3k
views
Trace inequality for non-reversible Markov chain
Let $P \in \mathbb{R}^{d \times d}$ be the transition kernel for a Markov chain with stationary measure $\pi$ and define $P^\ast$ to be the time-reversed transition kernel defined by $P^\ast_{ij} := ...
- 101
6
votes
2
answers
647
views
Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?
I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this ...
- 79
-2
votes
1
answer
309
views
Follow-up question re: logarithms of matrix-valued functions known not to have any zero eigenvalues [closed]
Given two parametrised "well-behaved" real skew-symmetric matrix-valued functions (of real variable(s)) $K(x)$ and $L(x)$, is it true that there exists another matrix-valued function $M(x)$ ...
- 286
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
0
votes
0
answers
107
views
Evaluating a matrix Pick function via its integral representation
In the proof of Theorem 3.1 of the paper Inequalities for M-matrices, Ando evaluates a matrix function (see equation boxed in orange below) via an integral representation of a Pick function (see ...
- 1,719
4
votes
1
answer
302
views
Hattori-Stallings trace
Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
- 985
3
votes
1
answer
130
views
Calculate the Riemannian Hessian of Karcher mean problem on positive definite matrices
Consider a collection of positive definite matrices $\{A_1,...,A_n\}\in\mathbb{S}_{++}^d$, the Karcher mean of these matrices is given by (see (5.4) in [1]):
$$
\min_{X\in\mathbb{S}_{++}^d} f(X):=\...
- 125
12
votes
0
answers
764
views
Does this matrix norm inequality have interesting application in other areas of mathematics?
In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices:
Theorem 3. Let $A=[a_{ij}]$ be a real symmetric ...
- 4,484
1
vote
1
answer
118
views
A sine type Chebyshev system
A sequence of real functions $\{\phi_1,\cdots,\phi_n\}$ is called a Chebyshev system on an interval $I\subseteq\mathbb{R}$, if any real linear combination $\sum_{l=1}^n a_l\phi_l$ has at most $n-1$ ...
- 4,058