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 = […]

\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 $\ rank(X)= \ rank(A)+\ rank(B)$.

$A$ 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 $X$ 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 […]

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 […]

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 […]

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 […]

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 $\det(\mathbf{M}) = (-1)^k \det(\mathbf{A}) \det(\mathbf{B})$?

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 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 […]

Intereting Posts

Compositions of $n$ with largest part at most $m$
Which functions on N extend uniquely to a continuous function on the Stone-Cech Compactification of N?
Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$
Escaping Gödel's proof
Am I fit for higher studies/teaching in mathematics?
Compute $P(X_1+\cdots+X_k\lt 1)$ for $(X_i)$ i.i.d. uniform on $(0,1)$
A(nother ignorant) question on phantom maps
satisfy the Euler-Lagrange equation
Counterexample of polynomials in infinite dimensional Banach spaces
Does using an ellipse as a template still produce an ellipse?
$\mathbb{Z}/m\mathbb{Z}$ is free when considered as a module over itself, but not free over $\mathbb{Z}$.
Why is epsilon not a rational number?
Find a branch for $(4+z^2)^{1/2}$ such that it is analytic in the complex plane slit along the imaginary axis from $-2i$ to $2i$
How to solve $ 227x \equiv 1 ~ (\text{mod} ~ 2011) $?
How to prove that the set $\{\sin(x),\sin(2x),…,\sin(mx)\}$ is linearly independent?