Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
986
questions with no upvoted or accepted answers
11
votes
0
answers
197
views
Expressing diagonal matrix using elementary matrices as generators in $\operatorname{SL}(\mathcal O_K \oplus \mathfrak a)$
Let $K$ be a real quadratic number field, $\mathcal O_K$ its ring of integers and an $\mathfrak a \subset K$ a fractional ideal. I've read in van der Geer that the group
$$\operatorname{SL}(\mathcal ...
- 6,184
10
votes
0
answers
351
views
Matrix powers of product of diagonalizable and orthogonal matrix
Suppose I have the following matrix constructed from some orthogonal matrix $O$ and a $\pm 1$ diagonal matrix $D=diag(\pm1,\dots,\pm1)$
$$
A = O D O^{-1} D.
$$
Is there a simple way to evaluate $A^n$ ...
- 129
9
votes
0
answers
2k
views
Can I go from the LU factorization of a symmetric matrix to its Cholesky factorization, without starting over?
I mistakenly computed the LU factorization and then realized that the question is asking for a Cholesky factorization, i.e., finding a lower triangular matrix L such that the symmetric matrix A has ...
- 1
8
votes
1
answer
216
views
Triangularization of matrix over PID
Let $R$ be a PID and let $A \in Mat_n(R)$ with all of its eigenvalues in R. Is it true that I can always find $P \in GL_n(R)$ such that $P^{-1}AP$ is uppertriangular? If so can I have a reference ...
- 81
8
votes
0
answers
11k
views
Decompose 3D rotation matrix into rotation around x, y and z-axis
I have a rotation matrix R, that produces an arbitrary rotation in a 3D space.
I would like to decompose it into 3 rotation matrices Rx, Ry and Rz so I can use and apply only xy in plane rotation (...
7
votes
0
answers
366
views
Fast arbitrary decomposition of a positive-definite matrix
Given a positive-definite $n\times n$ matrix $\mathbf{A}$, my goal is to present it as a product of the form $\mathbf{H^TH}$, where $\mathbf{H}$ is an arbitrary $n\times n$ matrix.
Cholesky ...
- 4,123
7
votes
1
answer
232
views
Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$
Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal.
I am trying to simplify the following expression
\begin{align}
{\rm Tr} \left( ...
- 6,015
7
votes
0
answers
659
views
Cholesky decomposition of $A+kI$ given Cholesky decomposition of A
Suppose I have the Cholesky decomposition for a symmetric matrix $A$:
$$
A = L L^T
$$
I wish to compute the Cholesky decomposition for $A+kI$ where $I$ is the identity and $k$ is a scalar. Is there ...
- 443
6
votes
0
answers
2k
views
Convergence of the complex QR algorithm to Schur decomposition
I study the complex Schur decomposition of a complex matrix $A \in \mathbb{C}^{n \times n}$, that is:
$$ A = U T U^H $$
where $T$ is upper-triangular (the eigenvalues of $A$ appear on its diagonal, ...
- 332
5
votes
0
answers
217
views
Dimension free gram matrix inner product
Let $\{x_i\}_{i = 1}^n$ be $n$ vectors of $d$ dimesnions.
We stack each $x_i$ as a row vector to form a matrix $X$ of dimension $\mathbb{R}^{n\times d}$
Let $\{y_i\}_{i = 1}^n$ be scalars (say all are ...
- 477
5
votes
0
answers
50
views
Difference of positive semi-definite matrices
If we have $S$ positive semi-definite matrices $A_1,\dots, A_S$ then what is the largest matrix positive semi definite matrix C such that $A_s -C$ is also psd for all $s=1,\dotsc,S$?
By largest I mean ...
5
votes
3
answers
5k
views
Huge matrix multiplication
I have a sparse A matrix stored in column major order (it is intrisically column major) of ~80GB and another sparse matrix B relatively small (1GB) which can be loaded in row or column major with no ...
5
votes
0
answers
574
views
What happens to woodbury matrix identity when A is not invertible?
The Woodbury matrix identity is
\begin{equation}
(A+UCV)^{-1}=A^{-1}-A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}.
\end{equation}
This formula suppose that $A$, $(A+UCV)$ and $(C^{-1}+VA^{-1}U)$
are ...
- 389
5
votes
0
answers
8k
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Checking positive semidefiniteness in MATLAB
Let $\mathbf{A}$ be a $n\times n$ matrix. I want to check in MATLAB if it is PSD or not. Which tests, in MATLAB, should I do for this purpose?
I know that if $\mathbf{A}$ is PSD then following holds
...
- 997
5
votes
0
answers
637
views
Is there a decomposition $U U^T$?
We know that there exist the Choleski decomposition
$ M = L L^T $
where $M$ is a positive definite matrix and $L$ a lower triangular one.
Does it exist a similar decomposition in
$ M = U U^T$
...
- 249