All Questions
Tagged with matrix-decomposition block-matrices
41
questions
1
vote
0
answers
27
views
A low-rank approximation problem with rank constraints
I am seeking a solution or some ideas to address the following problem:
$$
\begin{aligned}
&\text { minimize }_{\widehat{A}, \widehat{B}} \quad\|A-\widehat{A}\|_2 + \|B-\widehat{B}\|_2 \\ &\...
- 11
2
votes
1
answer
190
views
How to block diagonalize a orthogonal matrix?
I am trying to block diagonalize a real orthogonal matrix, A. The condition is that the blocks should also be orthogonal. I found this pretty old yet abstract paper that says
"By block ...
- 71
1
vote
0
answers
35
views
Decomposing a matrix $M$ in the form $M = P^{-1}QP$ where $Q$ and $P$ are real matrices and Q is as diagonal as possible
I am currently working on a tiny matrix library in C++ to help myself learn more about them. So far, I have implemented basic functions such as addition, subtraction, multiplication, the determinant, ...
- 19
2
votes
2
answers
90
views
Inverse of $3\times3$ block upper triangle matrix
How to find the inverse of $3\times 3$ block upper triangular matrix
$$X = \begin{bmatrix}
\mathbb{1} & \mathbb{B} & 0\\
0 & \mathbb{1} & \mathbb{B}\\
0 & 0 & \mathbb{1}
\end{...
- 151
2
votes
1
answer
182
views
SVD of complex matrix and real-valued representation
Consider a matrix $A \in \mathbb{C}^{m\times n}$. The SVD of $A$ reads:
\begin{equation}
A = U\Sigma V^H
\end{equation}
where $U \in \mathbb{C}^{m\times m},V \in\mathbb{C}^{n\times n}, \Sigma \in\...
- 127
0
votes
0
answers
49
views
Inverse of part of block matrix
I am trying to write a proof and, in order to do so, I have to simplify the following
$$ \left( E H B^{-1} \right)^b \bigg(\big(V-VH(B+HVH)^{-1}HV\big)^{bb}\bigg)^{-1}$$
where $b$ indicates the rows ...
- 70
0
votes
0
answers
40
views
Compare the condition number for the least square problem matrix
I am working on a problem with the following matrix $$G=\begin{pmatrix}I&A\\
A^T&0 \end{pmatrix}$$ where $A\in\mathbb{R}^{m\times n}$ and $m>n$ with full column rank.
Then by rescaling, ...
- 380
1
vote
2
answers
131
views
Can I decompose this matrix into separate parts?
Right now I have a matrix of the form
\begin{pmatrix}
0 & (a^\top b) ~ b ~c^\top \\
c ~ b^\top (b^\top a) & 0,
\end{pmatrix}
where $a$ and $b$ are vectors of the same dimension, and $c$ is ...
- 153
1
vote
0
answers
68
views
Eigenvalue bounds on $2\times 2$ block matrix with diagonal off-diagonals
Let $A,D\in\mathbb{C}^{n\times n}$ with eigenvalues $\alpha_i,\delta_i$ and let $b,c\in\mathbb{C}$. Without loss of generality, I've been able to prove the following statements.
The eigenvalues of $K ...
- 61
2
votes
1
answer
516
views
Block-diagonalization with unitary similarity transformations: ($A \rightarrow U B U^\dagger $, $B$ block-diagonal)
The problem
Given a matrix $A$, find a unitary matrix $U$ such that $U^\dagger A U=B$, where $B$ is approximately block-diagonal (when possible).
Explanation
In other words, assume I'm given a matrix ...
- 204
2
votes
1
answer
113
views
Decompose matrix in blocks with det = 0
I have a linear system which I know $\left[A\right]$ and $\left[B\right]$ and want to find $\left[X\right]$
$$
\left[A\right]_{(n+m) \times(n+m) } \cdot \left[X\right]_{n+m} =
\left[B\right]_{n+m }
$...
- 1,300
2
votes
0
answers
93
views
Conditions under which a non-symmetric block matrix is diagonalisable
I have a $(2n)\times (2n)$ matrix defined in blocks:
$$
\begin{equation}
\begin{split}
M=\left[
\begin{array}{c|c}
A & B\\
\hline
C & D \\
\end{array}
\right]
\end{split}
\end{equation}
$$
...
- 299
0
votes
0
answers
358
views
MATLAB Code Help. Using Crout's Method, solve the system of linear equation $Mz=f$, where $M=\begin{pmatrix}I &A\\A^T&0\end{pmatrix}$
Using Crout's Method, solve the system of linear equation $Mz=f$, where $$M=\begin{pmatrix}I &A\\A^T&0\end{pmatrix}$$
I have implemented algorithm of Crout's method. But I don't have any idea ...
1
vote
1
answer
165
views
Transform stacked matrix into block-diagonal form
Consider two matrices $A$ and $B$ that get stacked to form a (tall) matrix $J$,
$$
J = \left[\begin{array}{l}
A\\
B
\end{array}
\right].
$$
Assume that $\text{rank}(J) = \text{rank}(A) + \text{rank}(B)...
- 195
2
votes
1
answer
180
views
Eigenvalues of a blockmatrix
I am very interested in finding the eigenvalues of the matrix $\bf{A}$ below, which consists of $4$ block matrices of the same size $n \times n$ ,
$
\textbf{A}=\begin{pmatrix} {\bf{A}}_1 & {\bf{A}}...
- 151