Questions tagged [matrix-decomposition]

Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.

2,697 questions
Filter by
Sorted by
Tagged with
1 vote
16 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 ...
• 933
25 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
30 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
15 views

12 views

Norm inequality for eigendecomposition and oblique projectors

Consider a stochastic matrix R which permits an eigendecomposition into oblique projectors, $$R = \sum_{\lambda} \lambda C_{\lambda}$$ I've observed the following 2-norm inequality in a number of $R$ ...
• 21
22 views

Inverting a specific symmetric matrix preserves its zero entries

Suppose $S$ is an invertible symmetric matrix with the following property: For the entry in the $i$th row and $j$th column, if $|i-j|$ is an odd number then $S_{ij} = 0$; if $|i-j|$ is an even number ...
• 225
23 views

Is polar decomposition commutative for diagonal matrices?

I did a error while understanding about the polar decompositon. I thought polar decomposition is PU, but it is UP. While trying ...
• 111
36 views

1 vote
34 views

• 29
1 vote
52 views

What is a complete geometric interpretation of the eigendecomposition of matrices?

For reference, eigendecomposition of a matrix $A$ $\in R^{n \times n}$ is defined as: $A = P \Lambda P^{-1}$ where $P$ is a matrix whose columns are the eigenvectors of $A$, and $\Lambda$ is a ...
13 views

The order of decomposition of matrices

In QR decomposition of a matrix, Q represents the orthogonal matrix and R represent upper triangular. I was wondering if a matrix could be decomposed into orthogonal and lower triangular and found LQ ...
• 43
29 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 ...
• 286
38 views

find $D$ of $(D+A)$ for $diag((D+A)^-1)=k$

I am wondering whether it is possible to derive $D$ for $diag((D+A)^{-1})=k$ where $diag()$ produces a vector of diagonal elements of a squared matrix, $D$ is an unknown diagonal matrix with possible ...
• 141
38 views

Solving Matrix Equation using SVD

I'm reading this paper by Bishop and Tipping. They solve the equation $$(SC^{-1} - I)W = 0$$ Where $W \in \mathbb{R}^{d \times q}$ and $S , C \in \mathbb{R}^{d \times d}$ and $C = WW^T + \sigma^2 I$ ...
• 31
1 vote
48 views

Estimating the parameters of an ellipse

Problem definition Consider a dataset composed by $m$ bivariate measurements \begin{equation*} y_j \sim \mathcal{U}(\mathcal{E}(\ell_1,\ell_2,\theta)) \qquad j=1,2,\dots,m \end{equation*} uniformly ...
• 699