Articles of block matrices

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked – I searched but couldn’t find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) $B$ that is block diagonal form: I know both of these. I’m looking for a similarity matrix $Q$ such that $B = […]

Prove that the rank of a block diagonal matrix equals the sum of the ranks of the matrices that are the main diagonal blocks.

\begin{equation*} X= \begin{pmatrix} A& 0 \newline 0& B \end{pmatrix} \end{equation*} If A and B are some matrices and 0 is a zero matrix, prove that $\ rank(X)=\ rank(A)+\ rank(B)$. Also, if the upper right zero matrix would be replaced with matrix C, would it still be true, that always $\ rank(X)= \ rank(A)+\ rank(B)$.

Is always possible to find a generalized eigenvector for the Jordan basis M?

$A$ is a defective matrix, meaning that there are fewer linearly independent eigenvectors than eigenvalues; the algebraic multiplicity of $\lambda_1$ is $v_i = 2$ while the geometric multiplicity is $\mu_1 = 1$: $$ A = \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}, \lambda_1 = 1, e_1 = \begin{bmatrix}1 \\0\end{bmatrix} $$ The block diagonal matrix $J$ […]

Given the inverse of a block matrix – Complete problem

Given $X$ a block matrix $$\pmatrix{A&B}$$ where $A$ is $m \times n$ and $B$ is $m \times (n−m)$. I know a priori the value of $X \times (X^{T} \times X)^{-1}$. Substituting $X$: $$\pmatrix{A&B} \times \pmatrix{A^{T}A&A^{T}B\\B^{T}A&B^{T}B}^{-1}$$ I want to obtain the value of $A \times (A^{T} \times A)^{-1}$. I have tried many articles and tools, for […]

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is that, I have $2$ layers of atoms where $A$ is connectivity within the layer $1$ itself and $B$ is connectivity between layer […]

Sum of elements of inverse matrix

Assume NxN matrix A of complex values. I want to calculated the sum of all elements of its inverse. Does anybody have any good idea how to do this? The problem is that calculating the inverse is computationally expensive and since I am looking only for the sum of its elements, I thought there might […]

Simple to state yet tricky question

Define $$A=\left[\mathrm I+\sum_{k=1}^{m_1}v_k v_k^T+\sum_{k=1}^{m_2}u_k u_k^T\right]^{-1},$$ where each $u_k$ and $v_k$ is a $0$-$1$ column vector, and for each $1\leq i \leq n$, the $i$th component is $1$ in exactly one of $u$ vectors and in exactly one of $v$ vectors. Prove that for every $1\leq k \leq m$ if the $i’th$ element of $u_k$ is […]

How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix $$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$ How to prove that $\det(\mathbf{M}) = (-1)^k \det(\mathbf{A}) \det(\mathbf{B})$?

What is the codimension of matrices of rank $r$ as a manifold?

I’m reading through G&P’s Differential Topology book, but I hit a wall at the end of section 4. There is a result stating The set $X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$ is a submanifold of $\mathbb{R}^{m\times n}$ with codimension $(m-r)(n-r)$. There is a suggestion: Let $A\in M_{m\times n}(\mathbb{R})$ have form $$ A=\begin{pmatrix} B & C \\ D & […]

Show that the determinant of $A$ is equal to the product of its eigenvalues.

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I’m having a tough time figuring this one out. I know that I have to work with the characteristic polynomial of the matrix $\det(A-\lambda I)$. But when considering an $n \times n$ matrix I do not know […]