All Questions
6,026 questions
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votes
1
answer
327
views
A Matrix equation
Let $A$ and $B$ be two $n \times n$ full-rank matrices.
Let $XAY = B$ be the given equation where $X$ and $Y$ are unknown $n \times n$ matrices. We know that $Vec(B) = (Y^{T} \otimes X)Vec(A)$. Under ...
- 800
-2
votes
1
answer
470
views
Little conjecture about sums of reciprocals
Given a finite list $x_i$ of $N$ positive reals, it seems that $\sum_{i=1}^N x_i = \sum_{i=1}^N x_i {}^{-1} \Rightarrow \sum_{i=1}^N x_i \geq N$. Can anyone give me a proof?
- 2,513
-2
votes
1
answer
1k
views
Derivative of log determinant [closed]
Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative?
$$
\frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right).
$$
- 245
-2
votes
1
answer
213
views
Solving a difficult equation for a variable?
I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...
-2
votes
1
answer
162
views
What is the weakest condition on the matrices A_k that guarantees v_k->0 => A_kv_k->0 ? [closed]
What is the weakest condition on the sequence of real matrices A_k that guarantees that whenever a sequence of real vectores v_k converges to zero, the product A_kv_k also converges to zero?
Edit: ...
- 1,638
-2
votes
1
answer
475
views
sum of positive definite matrix
sum of positive definite matrix $A+B $is positive definite. I want to look at the spectrum of $C=A+B$
can we say the ith largest eigenvalue of $C$ is no less than the ith largest eigenvalue of $A$ i....
- 553
-2
votes
1
answer
968
views
What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]
Suppose we have the following symmetric matrix.
$$A = \sigma^2 I + u u^T$$
What can we say about the eigendecomposition of $A$?
- 67
-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
-2
votes
1
answer
183
views
Property of positive semi-definite
Let $A$ is a positive semi-definite matrix like this:
$$ A = \begin{bmatrix}
1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\
\alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\
\...
- 25
-2
votes
1
answer
353
views
Can we attain the maximum and minimum of a Rayleigh quotient over any subspace? [closed]
Let $M\in\mathbb{C}^{n\times n}$ be a Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
$$\mbox{Are } \sup_{x\in E\\
x\neq0}\dfrac{x^*Mx}{x^*x}\mbox{ and }\inf_{x\in E\\
x\neq0}\dfrac{x^*...
- 596
-2
votes
1
answer
140
views
Find a columns of matrix $A$ which form a basis of columns space of matrix $A$ [closed]
We have a matrix $A$ whose rows are data records and whose columns are features. We would like to omit useless features such as zero or constant columns, duplicate columns, columns that are equal to ...
- 1
-2
votes
1
answer
48
views
Rotating a known vector over two axis-es to result to another known vector [closed]
Lets assume i have a known vector, for example x = [1,0,0]
After 2 rotations, one over the y axis and one over the z axis, i result in a vector which in this example is x' = [0.5774, 0.5774, 0.5774]
...
-2
votes
1
answer
871
views
Rank of a random matrix
Let $x$ a random Gaussian vector of size $n$ with i.i.d coefficients $N(0,1)$. Let $J$ a random matrix with i.i.d coefficients $N(0,\sigma^2/n)$ where $\sigma \in [0,1]$. For any integer T>n, define:
$...
- 840
-3
votes
1
answer
3k
views
Are there infinitely many equivalence classes of similar matrices? [closed]
It is easy to show that similarity in matrices is an equivalence relation (two matrices A and B of same size being similar if there exists a matrix P such that B = PAP^(-1) )
Moreover, given a matrix, ...
- 2,855
-3
votes
1
answer
375
views
Opposite complex structure on Kaehler manifold
Let $(M,J)$ be a Kaehler manifold. How can one describe the opposite complex structure? What is the precise definition of the opposite complex structure? Can one describe the opposite complex ...
- 11
-3
votes
1
answer
123
views
Are the first 4 statistical moments independent? [closed]
Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...
- 11
-3
votes
1
answer
232
views
A problem that involves matrix and Lorentz Transformation [closed]
To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes.
$1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G A^T=A^...
- 45
-3
votes
1
answer
2k
views
Eliminating redundant linear constraints? [closed]
I have an NxN matrix of linear constraints that is not of full rank. In other words, some of the constraints are linear combinations of other constraints. The "standard" linear algebra tools (...
- 159
-3
votes
1
answer
167
views
Is there a simple function similar to exp? [closed]
As far as I know exp have such properties:
$f'(x) >0$
$f''(x) >0$
$\lim_{x \to -\infty}f(x)=0$
$\lim_{x \to +\infty}f(x)=\infty$
$f(x)f(-x)=1$
Let's say f(x) comply such rules.
The closest I ...
- 1
-3
votes
0
answers
145
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
-3
votes
1
answer
134
views
SU(2) and entangled particles [closed]
We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle_A\otimes \left| ...
- 1
-4
votes
2
answers
6k
views
Factorizing polynomials of several variables (in a different perespective)
I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^...
-4
votes
1
answer
387
views
Eigenvalues of real symmetric matrix [closed]
Suppose $A$ is a $n \times n$ real symmetric matrix with entries $a_{ij}\geq 1 $ and $a_{ii} = 0 $. Is it possible to have sum of the absolute eigenvalues of
$A < 2 (n - 1).$
-4
votes
1
answer
293
views
How to calculate $y^T \mbox{diag}(A^T B A) \,y$ efficiently? [closed]
I want to calculate $$y^T \mbox{diag}(A^T B A) \,y$$ where
$y$ is a $n \times 1$ vector.
$A$ is a $m \times n$ matrix where $n \gg m$.
$B$ is a $m \times m$ symmetric positive definite matrix; the ...
- 95
-5
votes
1
answer
86
views
Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Using degenerate perturbation theory and ...
- 23
-9
votes
1
answer
338
views
Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?
Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true?
$\|A\|_{2}$ denotes ...
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