All Questions
Tagged with matrix-decomposition numerical-linear-algebra
228
questions
2
votes
0
answers
34
views
need a Jacobian of an orthonormal basis for the column space of a matrix $\mathbf{X}$
I require to find a linear decomposition of any tall matrix $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$, where $M \geq N$, into some uniquely defined decomposition $\mathbf{X}$ $=$ $\mathbf{A}$ $\...
- 41
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
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 ...
5
votes
1
answer
155
views
If I can solve $Ax = b$ efficiently, is there a method to solve $(I+A)x=b$ efficiently?
Say if I already have the LU factorization of a square matrix $A$, is there an efficient way to get the LU factorization of $I+A$? (We may assume all the matrix I mentioned is invertible.)
I know from ...
- 555
4
votes
1
answer
64
views
Binary matrix power for a specific entry.
Consider a square $n\times n$ matrix $A$ whose entries are binary, that is, for all $i, j\in [n]$, it holds that $A_{i, j} \in \{ 0, 1\}$.
I am interested in the following decision procedure:
Given a ...
- 141
0
votes
1
answer
33
views
Distance between subspaces with spectral norm
I was trying to prove this following theorem ,
Let
$$ W=\begin{bmatrix}
\underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2}
\end{bmatrix} $$
$...
0
votes
0
answers
38
views
Finding the eigenvalues of a tridiagonal block matrix of special form
Consider the following symmetric tridiagonal block matrix: $$\begin{bmatrix}
2I_{N \times N} & -I_{N \times N} & O & \dots & O &O \\
-I_{N \times N} & 2I_{N \times N} &...
- 292
1
vote
0
answers
109
views
Spectrum of compact self-adjoint operator vs symmetric matrix
I know that compact self-adjoint operators have a countable, orthonornal eigendecomposition, and the spectrum is real and positive, and the only cluster point is zero. I know that self-adjoint ...
- 2,039
2
votes
1
answer
17
views
Relation between eigenvalues of a general complex matrices and its realed one
Let $A=A_R+iA_I$ be a $n\times n$ complex matrix. If we want to solve a linear system with regard to $A$ and do not want to take complex arithmetic, then we often generate the following real matrix:
\...
- 21
0
votes
1
answer
35
views
When complete pivoting fails for a linear system
As I understand it, the following three pivoting techniques:
Partial pivoting (which exchanges rows determined by a sub-column search)
Rook pivoting (which exchanges rows or columns based on sub-row ...
- 296
1
vote
1
answer
70
views
Show that $U^* A U$ is upper triangular, with $A$ upper triangular, and $U$ unitary and lower-Hessenberg
Let $A \in \mathbb{C}^{n \times n}$ be upper triangular. Let $u$ be a unit-norm eigenvector of $A$ whose eigenvalue is $a_{11}$ (the top-left entry of $A$). Let $U \in Unitary(n)$ be a lower-...
- 637
0
votes
0
answers
26
views
Decomposing a complex square matrix into multiple square matrices
I am working on a problem in optics (called Multi-plane light conversion) where I am trying to find the decomposition of a target transformation (matrix) $\hat{T}\in\mathbb{C}^{(NxN)}$. The form of ...
- 1
0
votes
0
answers
32
views
Solve linear IV-GMM numerically
I'm interesting in solving a linear IV-GMM (see page 5 and 6 for background). The solution takes the form
$$
\hat{\beta} = (X'ZWZ'X)^{-1} X'ZWZ'y
$$
where $W$ is a positive definite weighting matrix ...
- 147
2
votes
1
answer
327
views
LU decomposition of banded matrix with partial pivoting
Disclaimer: I'm rusty as can be in this department.
I'm looking into how to implement a banded matrix LU decomposition with partial pivoting ($PA = LU$). So to start with I implemented regular matrix ...
- 123
0
votes
0
answers
40
views
How to approximately diagonalize a special symmetric hermitian matrix?
Given a hermitian matrix $H$ as follows:
\begin{equation}
H =
\begin{bmatrix}
H^1 & V^{12} \\
V^{21} & H^2
\end{bmatrix}.
\end{equation}
Here, $H^1,H^2\in\mathbb{C}^{N\times N}$ ...
- 1