0
votes
1answer
19 views

Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
2
votes
1answer
68 views

Matrix Norm Inequalities and Linear Least Squares

I am working through one of my universities old QUALS and came across the following problem: Let $A,E\in \mathbb{C}^{m\times m}$. Suppose that $\sigma_\min>0$ is the smallest singular value of ...
3
votes
1answer
95 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
0
votes
1answer
33 views

Induced Matrix Norm

I have trouble following a proof of the induced Norm $||\cdot||_1$ The proof can be found here: ...
0
votes
3answers
62 views

About matrix products $A^{T}A$ and $ AA^{T} $

I'm investigating the relationship between 2-norms and eigenvalues of $A^{T}A$ and $ AA^{T} $, in order to better understand the SVD decomposition. How can I prove that $A^{T}A$ and $ AA^{T} $ are ...
1
vote
1answer
45 views

Special Matrix 2-norm and F-norm Inequalities

This is a homework problem for my Numerical Linear Algebra course. It states the following: If A is an mxm nonsingular matrix, prove the following: (1)$\|A+(A^{*})^{-1}\| _{2} \ge 2$ ...
0
votes
1answer
39 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
2
votes
1answer
50 views

Show that every operator norm is consistent

Is the following a correct way to show that operator norms are consistent? $$ \|AB\|=\max_{Bx \ne 0}\frac{\|ABx\|_\alpha }{\|x\|_\alpha} =\max_{ Bx\ne 0}\frac{\|ABx\|_\alpha}{\|Bx\|_\alpha} ...
0
votes
1answer
46 views

Generalized minimal residuals: eigenvalues and sets of functions

Can someone help me on this exercise (2 parts)? Thanks! Suppose that $S \subseteq \mathbb{C}$ is a set whose convex hull contains $0$ in it's interior (so $S$ is contained in no half-plane ...
-3
votes
1answer
63 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
2
votes
1answer
251 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
0
votes
3answers
87 views

Proving nonsingularity of a sum of matrices

I'm trying to solve this study question but I'm not sure how to proceed. The question is as follows. If \begin{equation}\frac{||B||_2}{||A||_2}<\frac{1}{\kappa_2(A)}\end{equation} with ...
1
vote
2answers
104 views

Matrix norm less than $1$ iteration

Is the following true always for a matrix norm $$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$ Related to this given $r$ is positive constant, $H$ is symmetric positive ...
0
votes
1answer
178 views

Solution for a Frobenius norm inequality

Am trying to find a real scalar $\gamma$ such that for a given pair of real rectangular matrices $X,Y$ the following holds: $\frac{||Y||_{F}^{2}}{5} \leq ||\gamma X||_{F}^{2}\leq ||Y||_{F}^{2}$ ...
0
votes
1answer
76 views

Scalar multiplication and Frobenius norm

Was wondering on what would be the real number (scalar) $\gamma$ that needs to be multiplied with each entry in a real rectangular matrix $X_{m\times n}$ such that the Frobenius norm of $X$ equals a ...