An inequality from the “Interlacing-1” paper

This question is in reference to this paper, (or its arxiv version, For the argument to work one seems to need that for the symmetric ±1 signing adjacency matrix of the graph, As, it holds that, max−root(∑s∈{0,1}mdet(xI−As))≤max−root(Es∈{0,1}m[det(xI−As)]) But why should this inequality be true? and the argument works because the polynomial on the … Read more

Basin of Attraction

I have a function F which is defined as follows: F(x)=N∑i=1f(zTix) where zi are known m×1 vectors, x is an m×1 vector, and for t∈R, f(t)=11+e−t/σ with a known σ (a sigmoid function). For a local or global maximum of F(x), I would like to characterize the basin of attraction—the set of all starting points … Read more

How does scaling rows to sum to 1, of a positive matrix change the perron vector?

Reposting from due to lack of response. Let A be a N×N positive matrix such that Aij>0. By Perron-Frobenius theorem, there is a unique positive left eigenvector called Perron vector x corresponding to the largest eigenvalue and also it sums to 1. Call it x. Let D be a diagonal matrix such that B=DA … Read more

Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the matrices of determinant $\pm1$ which are proportional to $$A_1:=\left(\begin{array}{cc}0.85&0.04\\-0.04&0.85\end{array}\right),\quad A_2:=\left(\begin{array}{cc}0.20&-0.26\\0.23&0.22\end{array}\right),\quad A_3:=\left(\begin{array}{cc}-0.15&0.28\\0.26&0.24\end{array}\right).$$ In fact, our question is slightly more specific: we would like to know that … Read more

Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M – \mu R_k$ with $R_k$ equals to a hermitian matrix and $\mu$ some positive constant. Moreover, I know that the the entries … Read more

Hodge duality and the determinant of the product of two matrices

I stumbled onto the following identity, and I would like to know: Is it known by some name and are there some references I might cite (or is it actually too trivial to be mentioned anywhere)? Are there any slick proofs that come to mind (for example something like Will Jagy’s in this somehow related … Read more

Number of linearly independent subsets in arbitrary set

Consider n-dimensional vector space Fnp over finite field Fp and a function a:Fnp→C. Let F(a)=∑S⊂FnpS is linearly independent∏v∈Sa(v), G(a)=∑S⊂FnpS is linearly independent|S|=n∏v∈Sa(v). For example, if M is multiset of vectors and a(v) is number of vectors v in M, F(M) and G(M) are number of linearly independent subsets and bases in M, respectively. Do F and G … Read more

non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix A=(aij)i,j=1,…,d∈Rd×d with the following property: aii=−∑j≠iaij, i.e., the matrix is not only weak diagonal-dominant, but its rows also sum up to 0. Note that the matrix is not necessarily symmetric (otherwise positive semi-definiteness would follow immediately from the weak diagonal-dominance). Furthermore, the matrix has the property that there are no … Read more

A trivialization of an almost complex structure

Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves. Roughly, one takes a solution $ u $ of a certain PDE (a la Cauchy-Riemann) and tries to find a tranformation $ \Phi $ and a holomorphic $ \sigma $ with $ u … Read more

Matrices in SL(2,C)SL(2,\mathbb{C}) with characteristic polynomial defined over a subring

Let R⊂C be a subring, and let A,B∈SL(2,C) be matrices such that A,B,AB all have trace in R. For which R can we then deduce that A,B are simultaneously conjugate to a pair of matrices in SL(2,R)? In particular I’m wondering about R=Z,Q, or a general number field. (Each of A,B is conjugate to a … Read more