All Questions
Tagged with matrix-decomposition normed-spaces
40
questions
0
votes
1
answer
220
views
Relationship between the Frobenius Norm of AB and BA?
Assume $A \in \mathbb{R}^{n\times m}$ is a $n\times m$ matrix, $B\in \mathbb{R}^{m\times n}$ is a $m\times n$ matrix, $\|AB\|_F \neq \|BA\|_F$ is definitely true at the most cases, but is there any ...
0
votes
1
answer
335
views
bounds on norm of a vector
Let $x \in \mathbb{R}^n$, and $M \in \mathbb{R}^{n\times n}$ be a full rank square matrix. If it is known that $ \| M x\|_2 \leq c$, then what can be said about the upper bound of $\| x\|_2$, i.e., $\|...
- 1,722
2
votes
0
answers
79
views
For a matrix, is there a generalization for the argument of a complex number like there is for the norm?
Any square complex matrix can be written in a polar form R U, where U = exp(i T) is unitary and ...
0
votes
1
answer
217
views
Bound norm of a product of matrices
Let $x_1 \in \mathbb{R}^{q_1}$, $x_2 \in \mathbb{R}^{q_2}$ be row vectors. Denote the $[2 \times (q_1 + q_2)]$ dimensional matrix \begin{equation}
X = \begin{pmatrix} x_1 & 0 \\ 0 & x_2 \end{...
- 55
3
votes
1
answer
85
views
If $U^T U = I_k$, $V^T V = I_{d-k}$, and $U^T V \approx 0$, is $U U^T + V V^T \approx I_d$?
Let $k$ and $d$ be integers with $1 \le k \le d-1$. Let $U \in \mathbb{R}^{d \times k}$ satisfy $U^T U = I_k$. Let $V, \in \mathbb{R}^{d \times (d-k)}$ satisfy $V^T V = I_{d-k}$.
A basic fact from ...
- 915
2
votes
0
answers
83
views
Upper bound in bayesian regression setting
Let $y_i = x_i^\top \beta + \epsilon_i$, $i=1,\ldots,n$; where $\epsilon_i$ are i.i.d. following a distribution with mean zero and unit variance, i.e., $\epsilon_i \sim P_{\epsilon_i}(0,1)$, $i=1,\...
- 81
0
votes
1
answer
135
views
matrix product, norm
Let $M_1, M_2, M_3$ are $n\times n$ matrices with real entries, and modulus of the eigenvalues are strictly less than one. For $x,y$ are any $n\times 1$ vector, Could anyone tell me given a suitable ...
- 36.3k
1
vote
1
answer
220
views
How to solve $F''_{w_{ab}}$? I want to know the detailed calculation process of solving $F''_{w_{ab}}$.
There is a objective function:
$f(W)$ = $||W||_{2,1}$
For any element $w_{ab}$ in $W$, we apply $F_{w_{ab}}$ to denote the part of $f(W)$ which is only related to $w_{ab}$.
$F'_{w_{ab}}=(DW)_{ab}$
...
- 311
1
vote
0
answers
187
views
Similarity of the solutions
I am trying to solve the following rank-k decomposition $$\Sigma \approx W\Lambda W^T$$
where $\Sigma \in R^{n \times n}$ is a positive definite symmetric matrix having diagonal elements 1, $\Lambda \...
- 81
0
votes
0
answers
334
views
What is the relationship between $||.||_{max}$ and energy of a matrix?
I was interested to find a relationship between $||.||_{max}$, i.e. max-norm of a matrix with that of its energy. $||.||_{max}$ of a matrix is defined by the maximum entry in the matrix. Generally, ...
- 3
0
votes
0
answers
61
views
Is Frobenius norm constant?
I came across an article that makes me doubt what I have learnt so far.
For example, let us decompose matrix A into L and U (LU factorization)
$A=\left(
\begin{array}{cccc}
5 & 4 & 1 &...
- 99
4
votes
1
answer
687
views
Condition number bound using block LU factorization
Let $A \in \mathbb{C}^{n \times n}$ be invertible and $P$ be a permutation matrix such that
$$PA = \begin{pmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} $$
Where $A_{11} \in \...
- 151
1
vote
0
answers
660
views
Directly prove strictly column diagonally dominant matrix is nonsingular using matrix norms
It is the case that, if a matrix $A \in \mathbb{C}^{n \times n}$ is strictly column diagonally dominant, meaning
$$|a_{jj}|> \sum_{i \neq j}^{n}|a_{ij}|, \quad 1 \leq j \leq n $$
then it is ...
- 167
1
vote
1
answer
2k
views
Does SVD give the best rank 1 approximation with respect to the Frobenius norm, L2 norm, or both?
From what I've observed in practice the SVD gives the best rank 1 approximation with respect to the Frobenius norm. But from what I've heard from others, it also minimizes the distance to the L2 ...
- 1,593
1
vote
0
answers
282
views
L1 norm constraint on product of 2 matrix
I want to solve below minimization problem
\begin{equation*}
\begin{aligned}
& \underset{A, B}{\text{minimize}}
& & ||Y-AB^T -D||_F^2 \\
& \text{subject to}
&& |A_i|_1 \leq a,...
- 81