0
votes
0answers
29 views

Computing characteristic polynomial of tridiagonal block matrices

I want to compute the characteristic polynomial of symmetric matrices of the form \begin{bmatrix} A & U & & & 0\\ U & B & V & &\\ & V& C& W &\\ ...
1
vote
1answer
15 views

determinant of special structure block matrix

How do you compute the determinant of the block matrix: $$ M = \begin{bmatrix} A+B & A &A &A &... \\ A & A+B &A &A &... \\ A & A &A+B &A &... ...
1
vote
1answer
69 views

Does there exist $B$ for which $BB^T=I$?

My question is Does there exist a real matrix $B_{n\times m}$ with $m<n$ for which $BB^T=I_n$? Why do I need this? Suppose we are given a real matrix $Q_{m\times n}$ (again, with ...
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 ...
2
votes
1answer
45 views

Determinant of $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$

Calculate the determinant of the following matrix: $M \in M_{2n}(\mathbb{C})$ such that $$M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$$ I find that that $\det M = 2^n$ is that ...
2
votes
1answer
155 views

Determinant of block matrix with certain properties

So I have the following 2N $\times$ 2N block matrix $H=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ where each block in an N$\times$N matrix. Each block have the following ...
1
vote
2answers
233 views

Determinant of block matrices with non square matrices

Let $A$ be $m \times n$ matrix, and B be $n \times m$ matrix, then Show that $\det\begin{bmatrix}I_{n} & B\\ A & I_{m} \end{bmatrix}=\det\begin{bmatrix}I_{m} & A\\ B & I_{n} ...
4
votes
2answers
108 views

Determinants of Block Matricies

I read on wikipedia that $Det \begin{pmatrix} A & B\\B& A\end{pmatrix}$ is equal to $ Det(A+B)Det(A-B) $ if $A$ and $B$ commute. Does this hold true even if $ A $ and $ B$ are not ...
0
votes
1answer
75 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
192 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 ...
5
votes
5answers
364 views

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 ...
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} = ...
0
votes
2answers
173 views

Linear algebra: need help with proof

Can someone please help me with this proof. For $A,B$ ∈ $F^{n×n}$, show that $AB$ and $BA$ have the same characteristic polynomial.
0
votes
4answers
85 views

Determinant of a $4\times4$ matrice with one unknown?

I have to calculate the determinant of this matrice. I want to use the rule of sarrus, but this does only work with a $3\times3$ matrice: $$ A= \begin{bmatrix} 1 & -2 & -6 & u \\ ...
5
votes
3answers
583 views

Determinants of block matrices

Let $A,B \in \mathbb{R}^{n,n}$. Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$. Show that $\det(C) \in \mathbb{R}$ ...
2
votes
0answers
91 views

Determinant, number of non zero columns

Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty : Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square ...