Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
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How to calculate the cost of Cholesky decomposition?
The cost of Cholesky decomposition is $n^3/3$ flops (A is a $n \times n$ matrix). Could anyone show me some steps to get this number? Thank you very much.
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Matrix factorization
I'd like to factorize matrices as follows:
$$ \left(\begin{array}{cc}X_1&X_2\\X_3&X_4\end{array}\right) = \left(\begin{array}{cc}D_1&D_2\\D_3&D_4\end{array}\right)\left(\begin{array}{...
- 2,160
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Proof of uniqueness of LU factorization
The first question: what is the proof that LU factorization of matrix is unique? Or am I mistaken?
The second question is, how can theentries of L below the main diagonal be obtained from the matrix $...
- 101
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Triangularizing a Matrix
In Hoffman & Kunze, it is mentioned several times that any matrix with a minimal polynomial that splits over your field into linear factors is similar to an upper triangular matrix. A proof is ...
- 1,903
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Block-diagonalizing an antisymmetric matrix
I was wondering how to block-diagonalize a $10 \times 10$ antisymmetric matrix into block matrices along the diagonal. Can I just diagonalize each non-diagonal block?
Thanks!
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Cholesky decomposition for sparse matrix
I have a symmetric positive definite matrix that is composed of small block diagonal matrices. For example:
$$
M = \left[
\begin{array}{ccc}
\Sigma & \Psi & \Psi \\
\Psi & \Sigma & \...
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Computing the Smith Normal Form
Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix
$$R := \begin{bmatrix}
-6 & 111 & -36 & 6\\
5 & -672 & 210 & 74\\
0 & -255 &...
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Singular Value Decomposition for zero-diagonal symmetric matrix
Let's say a zero-diagonal $4\times4$ symmetric matrix,
$$
\begin{bmatrix}
0 & 1 & 3 & 3 \\ 1 & 0 & 3 & 3 \\ 3 & 3 & 0 & 1 \\ 3 & 3 & 1 & 0
\end{...
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Can QR decomposition be used for matrix inversion?
Is there any simple algorithm for matrix inversion (that can be implemented using C/C++)?
Can QR decomposition be used for matrix inversion? How?
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Is it true that any matrix can be decomposed into product of rotation, reflection, shear, scaling and projection matrices?
It seems to me that any linear transformation in ${\Bbb R}^{n \times m}$ is just a series of applications of rotation — actually i think any rotation can be achieved by applying two reflections, but ...
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Triangularization of real matrices
Let $A = [a_{ij}] \in \mathbb{K}^{n \times n}$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}$.
In the case $\mathbb{K} = \mathbb{C}$, for example, via the Jordan decomposition ...
- 4,462
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Modified Cholesky factorization and retrieving the usual LT matrix
I have been looking at the modified Cholesky decomposition suggested by the following paper: Schnabel and Eskow, A Revised Modified Cholesky Factorization Algorithm, SIAM J. Optim. 9, pp. 1135-1148 (...
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Relation between Cholesky and SVD
When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed).
Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
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