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Results on Boolean matrices

We can try to produce some of the basic theory of Boolean matrices here to see what makes sense and what does not. For this post, we do not lose much by generalizing to the Boolean algebras of the ...
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Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix

Defining the $2 \times 2$ transfer matrix \begin{align}\tag{1} Q = \begin{pmatrix} -\lambda & 1 \\ -1 & 0 \end{pmatrix}, \end{align} the characteristic polynomial (CP) of the $M \times M$ ...
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Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix

I don't know about a general form, but at least for each of the cases of dimension $6n-1$ or $6n+1$, $n\in \mathbb{N} $, there is an eigenvalue $-1$ or $1$, respectively. Denoting $v_3 \equiv 1,0,-1$, ...
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Counting number of linear transformation from a given subspace to another subspace of same dimension

Are you asking for linear transformations $F_2^n\to F_2^n$ that take $U$ to $W$, or just surjective linear transformations $U\to W$? If the latter, the number is $(q^k-1)(q^k-q)\cdots (q^k-q^{k-1})$: ...
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Counting number of linear transformation from a given subspace to another subspace of same dimension

Recall that we can define a linear map by just defining where we send the basis vectors. So, to find how many surjective linear maps there are, we need to see how many ways can we take the basis ...
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KL divergence between gaussian with uniform prior

$\newcommand{\R}{\mathbb R}\newcommand{\KL}{{\operatorname{KL}}}$For $j=1,2$, let $P_j:=N(\mu_j,I_d)$, where $\mu_2=\mu_1+v$ and $v$ is a unit vector. So, for the pdf's $p_j$ of $P_j$ we have \begin{...
1 vote

Calculation of solid angle for rectangle in 6DOF

You can do this without integration, by performing the following steps. Calculate the coordinates of the vertices of your rectangle. Let the vertices be $v_1,v_2,v_3,v_4$. Let your point "...
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Do subgradient inequalities hold for matrix convex functions?

Some further research has pointed out the answer as affirmative (though I will leave the question unanswered for now, as I'm still perplexed by Ando's rank restriction, which now seems unnecessary). ...
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Eigenvectors that are tensor products?

Your $f(x) := \langle x^{\otimes r}, A x^{\otimes r}\rangle$ is a homogeneous polynomial of degree $2r$ of $d$ variables, $q(x) := ||x||^2$ is a homogeneous quadratic polynomial, and you look for the ...
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Eigenvectors that are tensor products?

This is NP-hard already when $r = 2$. To see this, I will consider the problem of minimizing your function $f$ instead of maximizing it, but it's not too much work to flip things around to see that ...
3 votes

Eigenvectors that are tensor products?

I strongly doubt any efficient algorithm exists for even approximately finding the maximizer in general. With slightly different notation, your problem is the same as the problem of "tensor ...
3 votes
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Reducing $9\times9$ determinant to $3\times3$ determinant

I think the simplest way to reduce $$A=M-\omega I$$ to a $3\times 3$ matrix is to use the Schur complement with respect to the $(\bar 2,\bar 2)$-elements of $A$, \begin{align} C = A/A_{\bar 2,\bar 2} =...
1 vote
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Probability of accurate sparse recovery

A good starting point is "Mathematics of sparsity (and a few other things)" by E. Candes, or a book on compressed sensing such as "A Mathematical Introduction to Compressive Sensing&...
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Reducing $9\times9$ determinant to $3\times3$ determinant

The formula's in the OP contain an error: the $\omega$ in the denominator of $N_1$ and $N_2$ should be $\omega^2$, so $$N_1 = \frac{aa^t}{\omega^2}, N_2 =\frac{a^ta}{\omega^2}\operatorname{id}.$$ Then ...
1 vote
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A question about the sign of quadratic forms on nonnegative vectors

I think $$M=\pmatrix{5&-3&-3\cr-3&5&-3\cr-3&-3&5\cr}$$ is a counterexample. If $z=(a,b,0)$, then $z^tMz=5a^2-6ab+5b^2\ge0$ for all $a,b$ (and, by symmetry, the same should be ...
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One question on circulant $(-1,1)$-matrices

This is a question about a sequence $a(t)\in \{\pm 1\}$ of period $n$ with 2 level periodic autocorrelations, with the nontrivial autocorrelations identically equal to 1. All these problems have a ...
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3 votes
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Non-convex combination of vectors

Yes, either point $u_0$ belongs to the convex hull $H$ of the points $u_1,\dots,u_k$ (in which case $\alpha_i$ exist), or there exists a hyperplane separating $u_0$ and $H$, when we can take $v$ as a ...
1 vote

Finding an analytical upper bound on linear transform of matrix

A result assuming that $M$ is positive definite. By continuity, in the optimal $\alpha$ the matrix $M+\alpha D$ is singular; hence the result is the smallest zero of $f(\alpha) = \det (M + \alpha D)$, ...
1 vote

Determinant with factorials is not 0?

A direct proof of T. Amdeberhan's identity (1) is as follows: we have $(i+j+t)!=(i+t)! f_j(i)$, where $f_j(x)=(x+1)(x+2)\ldots (x+j)$. Thus $$ \det ((i+j+t)!)_{0\leqslant i,j\leqslant n-1}=\prod_{i=0}^...
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Companion matrices must have geometric multiplicity one, linear recurrence sequence view

If $M$ had more than one Jordan block corresponding to some eigenvalue, then its minimal polynomial's degree would be smaller than $k$. This yields that all sequences satisfying your recurrence ...
5 votes

In a set of n points on $R^d$, each point can be "well separated" from the rest by a linear functional. Is the dimension necessarily $\Omega(n)$?

Take all the vectors of the form $e_i+e_j$, $i\neq j$, where $e_1,\dots,e_d$ is a base in $\mathbb R^d$. They satisfy your requirements, the vector $e_i+e_j$ being separated from the others by $e_i^*+...
5 votes
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One question on block-circulant matrices

The formula for the specific case is $$\det K=\det(A+B+C+D)\det(A-B+C-D)\det(A+iB-C-iD)\det(A-iB-C+iD).$$ More generally, for a block-circulant matrix with $n$ square blocks $A_0,\ldots,A_{n-1}$, the ...
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6 votes

An elementary inequality of operators

Iosif's example can be given a more conceptual description. Take $a=P$, $b=Q$ as projections. Then $P^p=P$, $Q^p=Q$, so the desired inequality becomes $$ (P+Q)^p \le P+Q . $$ Now $T^p\le T$, for $0<...
5 votes

An elementary inequality of operators

No. E.g., (identifying linear operators with matrices in a standard manner) let $$a=\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} \right),\quad b=\frac12 \left( \begin{array}{cc} ...
1 vote

Is the affine geometry a geometry of proportions?

The midpoint of $a$ and $b$ can be defined from equiproportion as the unique $m$ for which $(a,m,b)$ and $(b,m,a)$ are equiproportional. Similarly, the collinearity of $a,b,c$ can be defined as one of ...
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6 votes
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Analytical form for the nuclear norm of an $n \times n$ matrix

As Gro-Tsen pointed out, I had not computed the Galois group of the governing polynomial, and, in fact, my original answer was wrong. I believe that the following answer is correct, though. If by '...

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