1
vote
1answer
20 views

The similarity of the block matrices

Let $\mathbb{F}$ be a field,and let $A,B,C$ be matrices over $\mathbb{F}$ of respective sizes $n\times n , k\times k, $and $n\times k$. put $M=\begin{bmatrix} A&0 \\ 0&B ...
0
votes
1answer
40 views

Solve Matrix: How many trips can be made?

We have the matrix: $$\mathbf{M}=\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatrix}$$ The matrix ...
3
votes
1answer
59 views

Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
3
votes
2answers
73 views

Completely baffled by this question involving putting matrices in matrices

This is homework, so only hints please. Let $A\in M_{m\times m}(\mathbb{R})$ , $B\in M_{n\times n}(\mathbb{R})$ . Suppose there exist orthogonal matrices $P$ and $Q$ such that ...
4
votes
1answer
388 views

Determinant of a $2 \times 2$ block matrix

$\textbf{Problem}$: Let a $2n \times 2n$ matrix be given in the form $M=\left[ {\begin{array}{cc} A & B \\ C & D \\ \end{array} } \right]$, where ...
0
votes
1answer
74 views

proof about deteminant of a complex linear transformation

say I have a linear space $V$ over $\Bbb C$ and a linear transformation $T:V \to V$ such that $T=A+iB$ where $A,B \in \Bbb R^{n \times n}$ I proved already that $T_\Bbb R = \begin{pmatrix} A & -B ...
1
vote
2answers
189 views

deteminant of a block skew-symmetric matrix

If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix} how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix} so the determinant is ...
3
votes
1answer
105 views

Fast way to calculate determinant for a block matrix

I have a block matrix $$Q_{(n+m-1)\times(n+m-1)} = \begin{pmatrix} A & -J\\-J^t & B \end{pmatrix}$$ where $$A_{(m-1)\times(m-1)} = n*I_{(m-1)\times(m-1)} \text{ and } B_{n\times n} = ...
1
vote
1answer
45 views

Find a complex matrix that its square is a sum of a scalar matrix and a jordan block

Attempting to solve a question from homework, relating to Jordan normal form: Let $a \in \mathbb{C}, N \in M_n^\mathbb{C}$. $N$ is nilpotent of index 3. Prove that the matrix $S \in ...
2
votes
1answer
297 views

Block diagonalizing a real matrix

I need to prove that for every real linear operator $T:\mathbb{R}^n\longrightarrow \mathbb{R}^n$, there exists an orthonormal basis of $\mathbb{R}^n$ such that the corresponding matrix is block ...
2
votes
3answers
1k views

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 ...
0
votes
1answer
123 views

show blockmatrix is invertible

Let $B \in \mathbb{R}^{n \times n}, C \in \mathbb{R}^{m \times n}, m \leq n$ and $\operatorname{rank} C = m$ Suppose for every $v \neq 0$ with $Cv=0$ it is $v^TBv > 0$. Show: then $A = ...
1
vote
0answers
112 views

How does adding extra row and column of ones affect a matrix's inverse?

I'm working on a homework problem, and I'm stuck. I guess my linear algebra still needs some work... I've arrived at $\mathbf{D}= \left[ \begin{matrix} \mathbf{C} & \mathbf{1}^T \\ \mathbf{1} ...
1
vote
2answers
299 views

rank one update

Given a matrix $X$, we can compute its matrix exponential $e^X$. Now one entry of $X$ (say $x_{i,j}$) is changed to $b$, the updated matrix is denoted by $X'$. My problem is how to compute $e^{X'}$ ...