All Questions
Tagged with matrix-decomposition numerical-methods
82
questions
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
0
votes
0
answers
59
views
QR algorithm fails to converge (bad shift?)
Problem: My code-based implementation of the implicit QR algorithm fails to converge for certain special cases, and it's because those cases have bad shift values.
What are those special cases: While ...
- 433
0
votes
0
answers
71
views
Eigenvalue decomposition for $A^TA$ for sparse A?
I have a sparse matrix $A \in \mathbb{R}^{n \times l^{2}}$, and I want to calculate the eigenvalue decomposition of $A^{\top}A$. Since $A^{\top}A$ is positive semidefinite, all the eigenvalues are non-...
- 1,722
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
0
votes
0
answers
62
views
Numerical linear algebra problem (QR decomposition)
Problem: Given the QR-decomposition of a rectangular matrix $A \in \mathbb{R}^{m \times n}$, where $m > n > 1$, find the QR-decomposition of a matrix $A_k \in \mathbb{R}^{m \times (n - 1)}$,
...
- 596
3
votes
1
answer
139
views
Form of Q in extended QR decomposition calculated with Householder reflections
Let $A = QR$ be the extended QR decomposition of matrix $A \in \mathbb{R}^{m \times n}$ which is calculated by using $n$ Householder reflections. Prove by construction that there exist an upper ...
- 525
1
vote
1
answer
95
views
How to efficiently compute an SVD decomposition with a generalized orthonormal condition?
A regular SVD decomposition of matrix $X\in\mathbb{R}^{n\times m}$ is
$$ X = UDV^\top, \qquad U\in\mathbb{R}^{n\times r},\ D\in\mathbb{R}^{r\times r},V \in\mathbb{R}^{m\times r},$$
where $U$ and $V$ ...
- 157
0
votes
1
answer
468
views
Efficient low-rank approximation of the covariance matrix
Suppose we have $n$ samples each containing $p$ features arranged into a matrix $X \in \mathbb{R}^{n \times p}$.
We focus on the high-dimensional setting where $p >> n$.
By definition, the ...
- 836
0
votes
0
answers
53
views
Show that a matrix has a Cholesky factorization providing that it can be written as a product of a matrix and its transpose [duplicate]
$A$ is an invertible real square matrix ($A \in \mathbb{M_{n}(\mathbb{R})}$ and $det(A) \neq 0$).
Let's consider another matrix $B \in \mathbb{M_{n}(\mathbb{R})}$ such that:
$$B = {}^\intercal A \cdot ...
2
votes
1
answer
531
views
How to understand QR decomposition? Compare the power method, QR decomposition for finding eigenvalues and Lyapunov exponents.
The numerical methods for finding (the largest) eigenvalues and (the largest) Lyapunov exponents (LEs) look similar.
The power method is to use the matrix $B$ repetitively to grow a vector $z$, and ...
- 1,595
0
votes
0
answers
103
views
On the numerical computation of eigenvalues and eigenvectors
Given a matrix $A$ of order $n$ with coefficients in $\mathbb{C}$, applying the shifted QR algorithm:
$$
\begin{aligned}
& (Q, R) = \text{qrfactor}(A - \omega\,I_n)\,; \\
& A = R\cdot Q + \...
- 249
0
votes
0
answers
285
views
Uniquness of a QR-Decomposition: Show that there exists an orthogonal diagonal matrix $ S \in \mathbb{R}^{n \times n} $
Let $ A=Q_{1} R_{1}=Q_{2} R_{2} $ be two $ Q R $-decompositions of a quadratic matrix $A \in \mathbb{R}^{n \times n} $ with full rank, i.e., $ \operatorname{rank}(A)=n $. This means $ Q_{1}, Q_{2} \in ...
- 762
0
votes
0
answers
212
views
What's the general form of the rotation matrices should be used in QR decomposition
Apply two iteration of the QR method to the matrix that was given
$$
A=\left[\begin{array}{lll}
3 & 1 & 0 \\
1 & 3 & 1 \\
0 & 1 & 3
\end{array}\right]
$$
Solution:
$$
P_1=\...
- 875
0
votes
0
answers
268
views
Implementing LU factorization with partial pivoting in C using only one matrix
I have designed the following C function in order to compute the PA = LU factorization, using only one matrix to store and compute the data:
...
- 195
0
votes
1
answer
48
views
Constructing a vector norm on $\mathbb{R}^n$ such that subordinate matrix norm equals the spectral radius
Statement of problem: "Let $A$ be square diagonalizable matrix. Constructing a vector norm on $\mathbb{R}^n$ such that subordinate matrix norm, $||A||=\max|\lambda_i|$"
I know that $A$ being ...
- 395