New answers tagged

1 vote

How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

Here I'll present a few remarks and one reasonably efficient (IMHO) algorithm. Unlike my previous answer the conclusions here are experimental more than theoretical. You are absolutely right that the ...
  • 54.3k
0 votes

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 ...
2 votes
Accepted

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 $M \times M$ ...
0 votes

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 $v_3 \equiv 1,0,-1$, ...
0 votes

How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

I'm still not sure whether a closed form solution exists. But I recognized that most implementations of SVD will be iterative (the simplest being power iterations), so the block-coordinate descent in ...
3 votes

How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

OK, sorry for the delay: yesterday was hectic and then I got distracted by another question. Here is a small piece of theory promised. First, let us notice that $\|L-SR\|_F=\|LR^*-S\|_F$ so, as you ...
  • 54.3k
3 votes
Accepted

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} =...
6 votes

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
Accepted

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 ...
5 votes
Accepted

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 ...
  • 8,981
1 vote
Accepted

Eigenvalues of directed graph with one outward edge for each vertex

Here is an alternative (more combinatorial) proof to the one linked to in my comment. Suppose that the digraph $D$ has a vertex of in-degree zero, which we may assume is vertex $1$. Then letting $\...
  • 11.3k
1 vote
Accepted

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
Accepted

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 ...
  • 48.5k
6 votes
Accepted

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 '...
2 votes
Accepted

Distribution of scaled Johnson-Lindenstrauss transforms

$\newcommand\ep\epsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$We have \begin{equation*} P((1-\ep)\|x\|\le\|Ax\|\le(1+\ep)\|x\|)\ge\de \tag{1}\label{1} \end{equation*} for some $\ep,\de$...

Top 50 recent answers are included