Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
2,713
questions
2
votes
1
answer
44
views
Singular value of a bidiagonal matrix?
Consider a $n\times n$ matrix:
\begin{equation}
X=\begin{bmatrix}
a &1-a & & \\
& a &1-a & \\
& & \ddots &\\
& & & 1-a\\
& & & a
\end{...
- 304
-2
votes
1
answer
81
views
What is the square root of a square matrix squared? [closed]
Admittedly, made the title a little funny, but this is a valid question.
I have come across the following equation
$$
I x^2=AA
$$
where $I$ is a unit matrix, $A$ is a square matrix of the same ...
- 101
0
votes
0
answers
52
views
Suitability of QR factorization for solving a ill-conditioned linear system.
I am trying to solve a linear system $Ax = b$ where
$$
A =\begin{bmatrix}
2& 9& 2& 1& 4& 1& 0& 0& 0 \\
9& 65& 9& 1& 4& 1& 0& 0&...
- 1,261
1
vote
0
answers
23
views
Compressed image using SVD draws a clear line between part that's blank and part with a drawing. Why? [closed]
I'm trying to compress grayscale images using SVD. This is the original image:
Yes, there's a lot of blank space.
I then choose the x% largest singular values, perform the transformed matrices ...
0
votes
1
answer
23
views
For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric?
Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix $A\in\mathbb{R}^{n\times n}$ with ${\rm rank}(A)=r$, suppose the rank decomposition of $A$ is given as follows:
$$...
- 155
0
votes
0
answers
25
views
Singular values on streching the vectors
For the following statement: for a vector $x$ and a matrix $A$, if the vector $x$ is not in the null space of $A$, the vector $x$ will at least be stretched by the smallest non-zero singular value, i....
- 115
2
votes
0
answers
26
views
For a symmetric matrix $B$ and following four relevant matrices $P,Q,C,D$, what's the relation between $QP$ and $CDC^{\rm T}$?
Suppose $B\in\mathbb{R}^{n\times n}$ is a symmetric matrix with ${\rm rank}(B)=r$. Then $B$ is equivalent to $\tilde{B}$ in (1), where $I_{r}$ denotes the identity matrix of order $r$. That is, there ...
- 155
4
votes
1
answer
90
views
How to decompose a simple $3\times3$ shear transformation into a rotation, scale, and rotation
Is there a simple way to decompose the following $3\times3$ shear matrix into the product of a
rotation, (non-uniform) scale, and another rotation? Or Perhaps
some other combination of rotations and ...
- 802
0
votes
0
answers
72
views
Find one quartic root of a matrix
I have found the previous spectral decomposition of the matrix $$A=\begin{pmatrix} 1 & 1 & 0 \\
0&1&1\\
1&0&1
\end{pmatrix}.$$
You can see I verified such decomposition indeed ...
- 1,494
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:
$$
\begin{aligned}
&\text { minimize }_{\widehat{A}, \widehat{B}} \quad\|A-\widehat{A}\|_2 + \|B-\widehat{B}\|_2 \\ &\...
- 11
3
votes
1
answer
54
views
Completely non- normal matrix
Let $M_n(\mathbb{C})$ be the space of $n \times n$ matrices with complex entries. A matrix $N$ is said to be normal if $N^*N=NN^*$ where $N^*$ denotes the conjugate transpose of $N$. One can think of ...
- 973
0
votes
1
answer
34
views
Orthogonal projection of a complex valued matrix onto the space of Hermitian matrices
It is well known that any real matrix $A$ can be decomposed as the sum of a symmetric and a skew-symmetric matrix as follows:
$$
A= \frac{A+A^T}2+\frac{A-A^T}2.
$$
The decomposition is orthogonal, ...
- 141
0
votes
0
answers
34
views
Detect linearly dependent columns from a full-row rank matrix.
Let $A\in \mathbb{R}^{m\times n}$, with $m\leq n$, be a full-row-rank matrix. Then, there exist a collection of $n-m$ columns in $A$ that are linearly dependent on the other columns. What is the most ...
- 372
0
votes
0
answers
20
views
2x2 blocks in the QZ algorithm
How are the $2\times2$ blocks supposed to be diagonalized in the QZ-Algorithm? Taking the matrix pencil (A,B) and finding it's generalized Hessenberg decomposition (H,R) for which $\exists Q,Z \in \...
0
votes
1
answer
45
views
Matlab qz algorithm not reliable
I programmed my own version of the qz algorithm and, while testing it's results using the matlab qz algorithm, I found a particular case where my solution reaches an upper-triangular matrix and matlab ...