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Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following in …

Let $\{V_i\}_{i=1}^N$ be a set of $n\times m$, $n\geq m$, real matrices of full column rank and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. Moreover, let $A^{1/2}= …

Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following Lyapu …

Assume that $g(A,0)$ has $k$ distinct eigenvalues $\lambda_1,\dots,\lambda_k$. Then, does $g(A,t)$ have $k$ distinct eigenvalues for all (finite) values of $t> 0$? …

It is well-known that the eigenvalues of $A_n$ are $$ \text{eig}(A_n) =\left\{ a+2b\cos\left(\frac{\pi}{n+1}k\right), \ k=1,2,\dots,n \right\}. $$ My question. …