Tagged Questions
0
votes
1answer
25 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
56 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
60 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
35 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
77 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
169 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
71 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
81 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
191 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 $e^{X'}$ ...
