I ran into the following problem in my research: Given an invertible matrix $A \in \mathbb{C}^{n\times n}$, and define $B = A^{-1}$. Denote the $i$th standard basis in $\mathbb{R}^n$ by $e_i$ and the $i$th row of $B$ by $B_{i\bullet}$. Then is the following true \begin{equation} \label{1} \alpha A^HA – e_ie_i^T \succeq 0, \end{equation} when $\alpha […]

If $A$ and $B$ are $n \times n$ symmetric matrices with eigenvalues bigger or equal with $0$, how can I prove that $\det(A+B) \geq \det A +\det B$?

I have a $2\times2$ matrix $J$ of rank $2$ and a $2\times2$ diagonal positive definite matrix $Α$. Denote by $J^+$ a pseudoinverse of $J$. I can find many counterexamples for which $J^+AJ$ is not positive definite (e.g. $J=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)$ and $A=\left(\begin{smallmatrix}1&0\\0&2\end{smallmatrix}\right)$), but for all of them $J^+AJ$ has real and positive eigenvalues. So, I was wondering […]

Let $A \succ 0$ and $B$ be $n\times n$ symmetric matrices. Find the largest scalar $\gamma \geq 0$ such that $$A + \gamma B \succeq 0$$ Here is my (failed) attempt: The problem is: $$\max_{\gamma\ge 0} \gamma : v^\top(A+\gamma B)v \ge 0 \quad \forall v\in\mathbb R^n$$ Let $Au_i = \lambda u_i$ for $i=1,\dots,n$. And let […]

Let $\operatorname{Psym}_n$ be the cone of symmetric positive-definite matrices of size $n \times n$. How to prove the positive square root function $\sqrt{\cdot}:\operatorname{Psym}_n \to \operatorname{Psym}_n$ is uniformly continuous? I am quite sure this is true, since on any compact ball this clearly holds, and far enough from the origin, I think the rate of change […]

Let be $A$ a symmetric matrix in block form $$A= \begin{bmatrix} B & C \\ C^T & E\end{bmatrix}$$ and let $\operatorname{cs} A$ be the column space for $A$. Because $A$ is positive semidefinite $$\operatorname{cs} \left[\matrix{C^T & E}\right] = \operatorname{cs} E.$$ Why? Thanks for all explanations.

Let $0<\alpha<1$ and $A,B\in\mathbb{R}^{n\times n}$. I am trying to find conditions on $A$ and $B$ such that \begin{equation} I_n-\frac{1}{\alpha}B^{\rm T}B-\frac{1}{4\alpha(1-\alpha)}( A^{\rm T}B+A)^{\rm T}( A^{\rm T}B+A)>0. \end{equation} However, I do not know how to proceed. Any idea or suggestion is appreciated.

If $\mathbf M=\left[\begin{matrix}\bf A & \bf b\\\bf b^\top & \bf d \\\end{matrix}\right]$ such that $\bf A$ is positive definite, under what conditions is $\bf M$ positive definite, positive semidefinite and indefinite? It is readily seen that $\det(\mathbf M)=\alpha\det(\mathbf A)$, where $\alpha=\mathbf d-\bf b^\top A^{-1}b$ Now, $\alpha>0\Rightarrow \det(\mathbf M)>0$ $\quad(\det(\mathbf A)>0$ by hypothesis$)$ This is not […]

Assume that two symmetric positive definite matrices $X$ and $Y$ are such that $X-Y$ is a positive semidefinite matrix. Show that $$\det{(X)}\ge\det{(Y)}$$ I felt this result is clear, but I can’t use mathematics methods explain this why?

Let $A$ be a symmetric $n\times n$ matrix. I found a method on the web to check if $A$ is positive definite: $A$ is positive-definite if all the diagonal entries are positive, and each diagonal entry is greater than the sum of the absolute values of all other entries in the corresponding row/column. I couldn’t […]

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