All Questions
20
questions
1
vote
0
answers
54
views
Greatest lower bound and smallest upper bound of Frobenius norm of product of matrices
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, $\|A^{-1}\|.\|B\| < 1$ where $\|.\|$ is Frobenius norm.
I would like to find the lower bound and upper bound of $\|A^{-1}....
- 11
0
votes
1
answer
75
views
Lower bounding $v^TA^{\dagger}v$ where A is a sum of rank-1 matrices
This is related to a previous question I asked (Upper bounding $v^TAv$ where $A$ is the inverse of a sum of rank-$1$ matrices and $v$ is a vector).
Let $(x_i)_{1 \leq i \leq n}$ be vectors of $\...
- 192
1
vote
0
answers
485
views
Upper bound on the norm of an inverse of a positive definite matrix
Given two real-valued symmetric positive-definite matrices $A$ and $B$. Assume that $A\succeq I$. Let $\Delta = B-A$. We are interested in a bound on $\Vert B^{-1}-A^{-1}\Vert$ in terms of $\Vert \...
1
vote
1
answer
422
views
Upper Bound for the Induced 2-Norm $\|(A+B)^{-1}A\|$
Assume $A$ is a positive definite matrix, and $B$ is a positive semi-definite matrix. I am interested in the problem of whether there exists a constant upper bound for the induced 2-norm (spectral ...
- 163
-1
votes
1
answer
55
views
$\ell_2$ vs $\ell_{\infty}$ induced norm of a square matrix
I'm wondering if the result of this post $\| A \|_{L^2} \le \| A \|_{\infty}$ for symmetric matrices $A$ applies to square matrices that are not symmetric. Of course, for an asymmetrical square ...
- 155
3
votes
3
answers
363
views
Why $\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)}$
Given $A\succ 0$ (positive-definite) and $c>0$, I am trying to show
$$\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)} \tag{1}$$
using information like this but without success so far. Could ...
- 865
1
vote
1
answer
130
views
Bounding $\|A^n-B^n\|_F$ by $\|A-B\|_F$
Given
$$ \epsilon = \|A-B\|_F, $$
it is clear that
$$ \epsilon^n > \|(A-B)^n\|_F, $$
which follows from submultiplicativity.
I wonder if something related can be said about $\|A^n-B^n\|_F$?
- 195
0
votes
1
answer
53
views
Bound for $\Vert\sum_iA_i^\dagger A_i\Vert_\infty$ given that $\Vert\sum_i A_i\Vert_\infty$ is small
Let $A_i$ be matrices such that
$$\left\Vert \sum_i A_i \right\Vert_\infty \leq \varepsilon,$$
where $\Vert\cdot\Vert_\infty$ is the operator norm and is equal to the largest singular value of its ...
- 1,708
3
votes
2
answers
400
views
How to find upper and lower bound
Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
- 81
0
votes
0
answers
381
views
upper bound for quadratic form in terms of vector norm and eigenvalues
I have a quadratic form. if Q, P and M are positive and symmetric matrices.
$$(-x^T Q x - 2 x^T Q e - e^T Q e) + (y^T M y + 2 x^T P y + 2 e^T P y )$$
how can I get an upper bound for this quadratic ...
- 1
0
votes
1
answer
368
views
Relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$
Suppose we have two matrices $A\in \mathbb{R}^{m \times n}$ and $B\in \mathbb{R}^{n \times p}$, then what's the relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$? The notation $\|\cdot\|_*$ means ...
- 20
2
votes
1
answer
227
views
Show that $\|A\|_{\infty} \leq \sqrt n \|A\|_2$
Question:
Let $A \in \Bbb R^{m\times n}$. Show that $$\|A\|_{\infty} \leq \sqrt n \|A\|_2$$
Attempt:
First, I tried invoking the SVD (Singular Value Decomposition) of $A$:
$$\|A\|_\infty = \|UDV\|...
- 4,945
1
vote
0
answers
60
views
2-norm of product of multiple matrices
Suppose that $A \in \mathbb{R}^{m,n}$ and $B \in \mathbb{R}^{n,m}$ and $A$ has a bounded 2-norm. We know that $AB$ is a positive semi-definite matrix with $\|AB\|_2 \leq 1$. Further, assume that $C \...
- 1,180
3
votes
0
answers
244
views
Upper bound the maximum column sum of a particular stochastic matrix
Let $(x_i)_{i=1}^N$ be a set of vectors in $\mathbb{R}^D$. Define the matrix $W \in \mathbb{R}^{N \times N}$ as:
$W_{ij} = \frac{\exp(-||x_i-x_j||^2)}{\sum_k \exp(-||x_i-x_k||^2)}$
i.e. row $i$ of $...
- 51
1
vote
1
answer
859
views
A lower bound for the condition number matrix
I have the following proposition:
Theorem: For every invertible matrix $A\in\mathbb{R}^{n\times n}$ and every matrix norm $\|\cdot\|$, then the condition number $\mathcal{K}(A):=\|A\|\cdot\|A^{-1}\|$ ...
- 349