Lets say I have a block matrix of the form $ $ X = \begin{bmatrix} A & B & \ B^T & C \end{bmatrix}$ $ $ A, C$ , and $ X$ are all positive definite and I have bounds on both the minimum and maximum eigenvalue of $ A,C$ , and $ X$ .Read more
Lets say I have a block matrix of the form $ $ X = \begin{bmatrix} A & B & \ B^T & C \end{bmatrix}$ $ $ A, C$ , and $ X$ are all positive definite and I have bounds on both the minimum and maximum eigenvalue of $ A,C$ , and $ X$ .Read more
I’m trying to look at the evolution of the eigenvalues of a matrix as a function of magnetic field. My problem is that the eigenvectors are not ordered according to their eigenvector. I can sort these for a particular magnetic field value but I haven’t been able to extend this to include a table ofRead more
The following problem is motivated by one of my research problems. If you can solve this before me, I will include you as co-author in my work. Let $ \Sigma$ be an $ n \times n$ correlation matrix whose least eigenvalue is denoted by $ \lambda$ . $ \Sigma_i’$ be an $ (n-1) \times (n-1)$Read more
$ \newcommand\tr{\text{tr}} \newcommand\mean[1]{\left\langle{#1}\right\rangle}$ Let $ B\in\mathbb R^{n\times n}$ with its eigenvalues restricted to the left half plane, and anti-symmetric matrix $ T\in\mathbb R^{n\times n}$ satisfy $ $ B^\top-TB^\top=B+BT.$ $ Prove that $ \tr(TB)\leq0$ . What are necessary and sufficient conditions on $ B$ such that $ \tr(TB)=0$ . (e.g. it is sufficient for $ B$Read more
Let $ k|n$ and $ k\geq 2$ and consider $ k$ symmetric matrices $ $ A,B,\ldots,K \in \mathbb{R}^{n\times n}.$ $ Consider a $ k-$ interleaving of these matrices where we take every $ k^{th}$ row in order from the given matrices. I will illustrate with an example for $ k=2.$ Let $ $ A=\begin{bmatrix} a_0&Read more