All Questions
16
questions
0
votes
0
answers
23
views
The lower bound of Frobenius norm of matrices product.
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, where $\|.\|$ is Frobenius norm.
I would like to find the lower bound of $\|A^{-1}.B.(A+B)^{-1}\|$:
I have the result:
$$\|A^{...
- 11
0
votes
0
answers
23
views
How to find an upper bound for a norm with power less than 1?
I need to find an upper bound for $$\|x-y\|_2^\nu$$ in which $\nu \in [0,1]$. I was wondering if $\|x-y\|_2^2$ can be considered as an upper bound. or $\|x\|+\|y\|$?
1
vote
0
answers
485
views
Upper bound on the norm of an inverse of a positive definite matrix
Given two real-valued symmetric positive-definite matrices $A$ and $B$. Assume that $A\succeq I$. Let $\Delta = B-A$. We are interested in a bound on $\Vert B^{-1}-A^{-1}\Vert$ in terms of $\Vert \...
-1
votes
1
answer
55
views
$\ell_2$ vs $\ell_{\infty}$ induced norm of a square matrix
I'm wondering if the result of this post $\| A \|_{L^2} \le \| A \|_{\infty}$ for symmetric matrices $A$ applies to square matrices that are not symmetric. Of course, for an asymmetrical square ...
- 155
1
vote
1
answer
1k
views
Is the 2-norm of a matrix bounded by the maximum of its 1-norm and Infinity-norm?
I am implementing the algorithm in "Approximating the Logarithm of a Matrix to Specified Accuracy" by Sheung Hun Cheng, Nicholas J. Higham, Charles S. Kenny, Alan J. Laub, 2001.
In this ...
- 153
0
votes
1
answer
53
views
Bound for $\Vert\sum_iA_i^\dagger A_i\Vert_\infty$ given that $\Vert\sum_i A_i\Vert_\infty$ is small
Let $A_i$ be matrices such that
$$\left\Vert \sum_i A_i \right\Vert_\infty \leq \varepsilon,$$
where $\Vert\cdot\Vert_\infty$ is the operator norm and is equal to the largest singular value of its ...
- 1,708
3
votes
2
answers
400
views
How to find upper and lower bound
Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
- 81
1
vote
1
answer
93
views
Matrix norms translating with identity matrix
Let $\mathcal{A}$ denote a linear operator, $\mathcal{A}^*$ its adjoint, and $\text{I}$ the identity matrix. Is the norm of $\mathcal{A}$ related to the norm of $\mathcal{A}^* \mathcal{A} - \text{I}$?
...
- 13
0
votes
1
answer
368
views
Relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$
Suppose we have two matrices $A\in \mathbb{R}^{m \times n}$ and $B\in \mathbb{R}^{n \times p}$, then what's the relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$? The notation $\|\cdot\|_*$ means ...
- 20
4
votes
1
answer
59
views
After applying a sequence of involutory real matrices to a vector, is the norm of this vector bounded from below?
For $n,N \in \mathbb{N}$, let $A_1, \ldots, A_n$ be a finite sequence of involutory $(N \times N)$-matrices over $\mathbb{R}$, i.e. $A = A^{-1}$.
We know, that the eigenvalues of any involutory matrix ...
- 185
1
vote
0
answers
78
views
Trace and 2-norm of linear combination of outer products
Suppose that $c_i \in \mathbb{R}-\{0\}, B_i \in \mathbb{R}^{k \times m}, \alpha \in \mathbb{R}^k$ with $\|\alpha\|_2 = 1$. Consider the following linear combination of outer products:
$$M = \sum_{i=1}...
- 1,180
0
votes
1
answer
35
views
What conditions should vector $x$ satisfy so that $\|[x_2+\alpha x_1, \dots, x_n + \alpha x_1]\|_2$ is bounded by a constant?
Suppose that $x = [x_1,\dots,x_n]$ is a vector with norm less than or equal to one $\|x\|_2^2 \leq 1$. Let $\alpha \in [0,1]$ and define the following vector
$$y = [x_2+\alpha x_1, \dots, x_n + \...
- 1,180
1
vote
0
answers
60
views
2-norm of product of multiple matrices
Suppose that $A \in \mathbb{R}^{m,n}$ and $B \in \mathbb{R}^{n,m}$ and $A$ has a bounded 2-norm. We know that $AB$ is a positive semi-definite matrix with $\|AB\|_2 \leq 1$. Further, assume that $C \...
- 1,180
0
votes
1
answer
122
views
upper bound for the norm of vector belonging to the column space of a matrix
If $u$ is a vector that belong to the column space of a matrix $A$ and $\sigma_{\min}(A)$ is the smallest non zero eigenvalue of $A^T A$, then I read in a paper that we can write
$$ \sigma_{\min}(A)^{\...
- 105
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