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In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then t …

You conjecture is not true. Let $P$ be the $3\times 3$ matrix $$ \left( \begin{array}{ccc} 0&1&0\\ 0&0&1\\ 0&0&0\end{array} \right) $$ which is nilpotent with $P^2\neq 0$. Consider the subspace of $9 …

This came up in a conversation with an engineer friend of mine. Let $c>0$ be a constant. Let $A_{ij}$ be an $n$ by $n$ matrix with entries $$ A_{ij} = e^{-c(i-j)^2}. $$ Is there a name for this matri …

I don't know what's going on with the paper, but here is an argument regarding existence of such decompositions. Given a rank one decomposition $$X = \sum_{i=1}^R x_ix_i^T$$ one has $\sum_{i=1}^R x_ …

Clearly, the statement is invariant under multiplication by $a\in K$, so we may assume that $W\ni 1$. This implies that $W\supseteq R$, and we want to show that $W=S$. Suppose that $t\in W$. I claim …

This is not a complete solution by any means, but here are some ideas. If one of $A_j$ (or their linear combinations) is invertible, then one can get a necessary and sufficient condition. Namely, if …

For $\phi_j=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can factor out $\lambda_j-\lambda_j^{-1}$ to get a matrix $ …