I want to apply the Schur complement to one element of a block matrix. What I do not know is how to organize the resulting values inside the new matrix. E.g. Given the following block matrix with appropriate dimensions: \begin{equation} \begin{bmatrix} A-BD^{-1}B^T & E \\ E^T & C \end{bmatr...

Let $\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le \|A_0\|_2\}$ where $A_0 $ is some fixed matrix and $\|\cdot\|_2$ denotes the induced $2$-norm. We also have for every $A \in \mathcal E$, $\rho(A)< 1$ where $\rho(\cdot)$ denotes the spectral radius and $\rho(A_0) < 1$. Is i...

Let $\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le M \text{ and } \rho(A) < 1\}$ where $M \ge 1$ is some fixed constant and $\|\cdot\|_2$ denotes the induced $2$-norm. Is it possible to give an upper bound $C$ in terms of $M$ such that $\|(I-A)^{-1}\|_2 \le C$ for all $A \in...

Let $A \in \mathcal M(n \times n; \mathbb C)$ with $\rho(A) < 1$. Then we know $I \otimes I - A^T \otimes A^T$ is invertible. Let $\text{vec}$ denote the vectorization operation and $\mathcal T = (I \otimes I - A^T \otimes A^T)^{-1} : \mathbb C^{n^2} \to \mathbb C^{n^2}$. The operator norm of $\m...

Let $A \in \mathcal M(n \times n; \mathbb R)$ with $\rho(A) < 1$. Let $X, Y$ be the solutions to the following Lyapunov matrix equations \begin{align*} X &= A^T X A + Q \\ Y &= A Y A^T + Q \end{align*} where positive definite matrix $Q$ is given. By the assumptions on $A$, the solutions have cl...