All Questions
5,883 questions
0
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(probably simple) optimization question
Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
- 6,962
0
votes
2
answers
2k
views
How to accelerate/avoid multiplication for large matrices in Matlab? [closed]
The setting is here.
X: 6000x8000 non-sparse matrix
B: 8000x1 sparse vector with only tens of non-zeros
d: positive number
M: is sparsified X'X, i.e. thresholding the elements smaller than d ...
- 21
0
votes
2
answers
371
views
Quantum observables
Let H be a Hilbert space and A, B two non-commuting bounded linear operators. Let Com(A,B) be the set of bounded linear operators C which commute both with A and B.
Question 1 : What is known about ...
-1
votes
1
answer
681
views
Is there such a thing [closed]
For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le5$, are there
$W\in\mathcal{U}\left(4\right)$ and nontrivial $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$, such that $\mbox{tr}\left(U_{i}\mbox{...
- 53
-1
votes
1
answer
305
views
A simple matrix multiplication query [closed]
The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given ...
- 13.9k
-1
votes
2
answers
335
views
How to generate constant row and column sum matrices?
How can we randomly generate matrix $A \in \mathbb{R}^{n \times m}_{\geq 0}$ that satisfies
$A 1_n = m1$ and $A^T 1_m = n1$.
- 1
-1
votes
1
answer
809
views
On an eigenvalue inequality [closed]
Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| \...
- 29
-1
votes
2
answers
806
views
The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
- 251
-1
votes
1
answer
152
views
Topological characterization of invertible real matrices [closed]
Let $n\geq 2$ be an integer. Consider the topological space $M_n$ of $n$-by-$n$ matrices with real entries.
Can you give a short non-constructive proof of the existence of a continuous function $M_n\...
- 23
-1
votes
1
answer
185
views
eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $
Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
- 461
-1
votes
1
answer
247
views
Maximal commutative subrings of the endomorphism ring of a vector space
Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
- 1,959
-1
votes
1
answer
172
views
$A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?
My question follows from https://math.stackexchange.com/questions/3857976/inverse-inequality-of-symmetric-matrix. Suppose we assume that $A$ and $B$ are two positive definite matrices with positive ...
- 154
-1
votes
1
answer
215
views
Dense linear span implies closed convex hull has non-empty interior
Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
- 5,405
-1
votes
1
answer
195
views
Determinant of $Z^TZ$ [closed]
If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
- 2,246
-1
votes
1
answer
121
views
On rank one torsion-free modules over local rings [closed]
Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes_A k \le 1$? If not, is there a known upper-bound ...
- 2,126
-1
votes
1
answer
1k
views
How to show the square root function of a positive semidefinite matrix is differentiable? [closed]
How to show the square root function of a positive semidefinite matrix is differentiable?
In this context PSD means symmetric PSD.
- 111
-1
votes
1
answer
142
views
Action of rotation group on Matrices [closed]
Is the following assertion true?
Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in SO(p,\...
- 601
-1
votes
1
answer
1k
views
Sum of two unitary matrix is equal to every matrix? [closed]
Let $R=M_{n}(Z_{2})$, can we write every matrices of $R$ as sum of two matrices of $GL_{n}(Z_{2})$?
- 461
-1
votes
1
answer
137
views
Does a half plane contain intersection of some other half planes? [closed]
I'm doing research in Optimization and I have found this obstacle in the way.
If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
- 19
-1
votes
2
answers
438
views
Are the coefficients of a linear combination of random vectors as random?
Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...
- 271
-1
votes
1
answer
360
views
Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]
Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
- 7
-1
votes
1
answer
155
views
Companion matrices must have geometric multiplicity one, linear recurrence sequence view [closed]
I posted this question on math stackexchange weeks ago, and it have not receive an answer yet after a bounty offer...
I've been recently playing around with the linear recurrence sequences. Consider ...
- 109
-1
votes
1
answer
323
views
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
votes
1
answer
62
views
Basic operation geometrical meaning [closed]
What is the geometrical meaning of doing
$x^TAx \;$?
$Ax \; $ is trivially "applying A to x", but then, what the multiplication for $x^T$ stands for?
-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 $E$) ...
- 2,118
-1
votes
1
answer
174
views
Question about proof of positive roots under reflection
Since I did not receive a lot of responses on Math Stack Exchange
I would like to repost this question here.
Let $(W, S)$ be a finite Coxeter system. Furthermore, let $V$ be a real vector space with ...
- 103
-1
votes
1
answer
85
views
Optimizing a non-operator bilinear product over a Hilbert space
Sorry to use bra-ket notation, but I studied Physics, not Math.
Consider a bilinear-product defined by
$\langle x'|F|x \rangle = \langle x'|F(x',x)|x \rangle = F(x',x)$
such that, in general,
$\...
-1
votes
1
answer
119
views
Is this operation on infinite matrices associative?
This is an additional question to that question, inspired by Leonid Petrov.
Let $\text{Mat}(\mathbb{N},\{0,1\})$ be the set of all maps $A:\mathbb{N}\times\mathbb{N}\to \{0,1\}$. We define a matrix ...
- 48.9k
-1
votes
1
answer
173
views
finding a unitary submatrix inside a random matrix
Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...
- 482
-1
votes
1
answer
83
views
Dimension of some ideal in the group ring Z/p[Z/p]
Let I be the augmentation ideal of the group ring Z/p[Z/p] and I^n denotes the ideal generated by all possible products of n elements from I.
Question: What is dimension of I^n as a vector subspace ...
- 745
-1
votes
1
answer
2k
views
Inequality between two matrices
Given a full rank $n \times m$ matrix $K$ with $m<n$ and an invertible symmetric matrix $J$. Let $A$ be a symmetric positive semi-definite $n \times n$ matrix such that
\begin{equation}
(K^T K)^{-...
- 133
-1
votes
2
answers
605
views
Approximating a subspace by sampling a base without replacement
Let $X$ be a $p \times n$ matrix, with $p > n$. Now, suppose I sample $m < n$ columns from $X$ at random, without replacement. I would like to characterize the distance between the subspace ...
- 461
-1
votes
1
answer
77
views
Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
- 287
-1
votes
1
answer
68
views
Bound for an expectation of random matrix with quantized random variable
Let random matrix $\mathbf{X} \in \mathbb{C^{\mathrm{m} \times \mathrm{n}}}$ and random vector $\mathbf{y} \in \mathbb{C^{\mathrm{m} \times 1}}$ are unknown distributed, but their covariance and ...
- 25
-1
votes
1
answer
825
views
How to calculate determinants of such types?
Consider next determinant that we want to expand around $h=1$
\begin{eqnarray}
Z_q \ = \ h^{N N_f} \ \ \left ( \prod_{n=1}^{N} \ \sum_{l_n=0}^{N_f -q} \ h^{2l_n+q} \ \binom{N_f}{l_n} \right ) \ \...
-1
votes
1
answer
65
views
A follow-up question in a proof in a paper on complete multipartite graphs
A follow-up question from the following article/paper:
"Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion"
by Shaowei Sun and Kinkar Chandra ...
- 199
-1
votes
1
answer
60
views
Linear operator over a simplex space in a multinomial distribution parameter estimation problem
This is actually a variant of a well-known problem of how the parameters of a multinomial distribution can be estimated by maximum likelihood, and this arises from a final year project I undertook ...
- 1
-1
votes
1
answer
330
views
Holder inequality for a general rectangular matrix
Let $A \in \mathbb{R}^{m\times n}$ and $p,q \in \mathbb{R}^{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$. I am interested to prove the following:
$$ \|A\|_{p}=\|A^T\|_q$$
I have tried using Holder ...
-1
votes
1
answer
94
views
How many hyper-rectangle-like objects are intersecting a hyperplane?
Let $A\in \mathbb R^{n\times n},\ b\in \mathbb R^n$ such that $\forall x\in \{-1,1\}^n : Ax\ne b$.
Let us denote: $S=\{x\in\mathbb R^n|Ax=b\}$ ('S' for solution set).
Is $\ \#\Big\{H\in\big\{ \{-1\},...
- 265
-1
votes
1
answer
397
views
Bound for psd-matrix weighted norm of two related vectors
Let two vectors, $\mathbf{x}, \mathbf{y}$ be related as: $0 \leq x_i \leq \lambda y_i$, for some $\lambda > 0$. That is, $\mathbf{x}$ is coodinate-wise dominated by a scaled version of $\mathbf{y}$....
- 75
-1
votes
1
answer
180
views
Orthogonal polynomials of the second kind
Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $L(x^n)= s_n, n\ge 0.$ Given a positive ...
- 1
-1
votes
1
answer
132
views
How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? [closed]
Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha \log(A)...
- 201
-1
votes
1
answer
460
views
Orthogonal decomposition of conditional expectations
Suppose I have a random variable $x$ and a set of conditional distributions on $x$. Here is an example where the conditionals are nested:
$$q_1 := E(x|y_1), \quad q_2 := E(x|y_1,y_2),\quad q_3 := E(x|...
- 427
-1
votes
1
answer
293
views
spectrum of a special class of tridiagonal matrices
Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e.,
$$\begin{bmatrix}...
- 15
-1
votes
1
answer
200
views
Collecting terms of a linear expression with nested sums and combinatorics in coefficients
I need to collect the $\Pr(\cdot)$ terms of the following expression:
$\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left(
1-\theta \right) }\right) ^{m}}\left[ \sum_{j=2}^{m-1}...
-1
votes
1
answer
1k
views
Rank of covariance matrix whose diagonal elements are same [closed]
Suppose A is a covariance matrix whose diagonal elements are same, i.e. $A_{1,1}=A_{2,2}=\cdots=A_{N,N}$, can we conclude that A is full rank?
Suppose the absolute values of the off-diagonal elements ...
-1
votes
1
answer
175
views
Regularized Gradient with respect to a matrix (with a specific structure)
Suppose we have a typical logdet function $\mathcal{L}$
$$
\mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q},
$$
where $\...
- 125
-1
votes
1
answer
492
views
Upper bound on iterations count for power iteration algorithm
I'm stuck trying to get upper bound on iterations count for power iteration algroithm for finding first eigenvalue of adjacency matrix $A$ given tolerance value. I've tried to figure something out ...
- 89
-1
votes
1
answer
2k
views
Absolute values and Frobenius norm [closed]
The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
- 1
-1
votes
1
answer
383
views
On the permanent dominance conjecture for symmetric group
The Lieb's permanent dominance conjecture states that the expression $$\frac{d_{\chi}^HA}{\chi(e)}\le per(A)$$ holds for all positive semidefinite matrices $A$, where $d_{\chi}^HA=\sum_\limits{\sigma\...
- 2,089