1
vote
1answer
26 views

Lipschitz continuity for generalized inverse matrix

Suppose $A$ and $B$ are full-rank and well-conditioned. Is Lipschitz continuity held for generalized inverse? $$\|A^+ - B^+\| \le \omega \|A-B\|,$$ for some $\omega > 0$, where the norm could ...
0
votes
0answers
22 views

Lipschitz continuity of inverse

Given a function f : $\mathbb{R}^n\to\mathbb{R}^m$, which is known to be Lipschitz continuous, can we say anything about the Lipschitz continuity of it's inverse function (in this case, the ...
7
votes
2answers
77 views

The map that sends $A$ to its greatest eigenvalue is continuous.

The map $f:S_n(\mathbb R)\to \mathbb R$ such that $f(M)$ is the greatest eigenvalue of $M$ is continuous ($S_n(\mathbb R)$ is the set of symmetric matrices) I need to prove this result in order ...
2
votes
2answers
65 views

Constructing a function similar to x^3 between [0,1]

I'm trying to construct a function $f$, in order to normalize a dataset(obviously where all the element come from $[0,1] \in \mathbb{R}$. The big picture is that the envisioned $f: [0,1] \rightarrow ...
1
vote
1answer
59 views

Proof that eigenvector corresponding to simple eigenvalue is continuous

Let $\lambda$ be a simple eigenvalue of $A \in L(C^n)$ and let $x$ be the corresponding eigenvector. Then for $E \in L(C^n)$, $A+E$ has an eigenvalue $\lambda(E)$ and an eigenvector $x(E)$ such that ...
1
vote
1answer
38 views

Continuity of bilinear maps

Given a vector space $V$ over $\mathbb{R}$ with a norm $||*|| $. Can $(x,y)\rightarrow(x+y)$ be an example of continous bilinear map, if yes, can you please exlain why? Definition of continuous ...
0
votes
1answer
23 views

Metric induced from norm

I was trying to understand the following: Every norm on $R^n$ is continuous (as a map from $R^n$ to $R$). Proof. We use the maximum metric on $R^n$: $ d(x, y) = \max{|x_j − y_j| : j ∈ \{1, . . . ...
3
votes
0answers
48 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
1
vote
4answers
42 views

$M \rightarrow M^T M$ is a continuous mapping.

Consider the following mapping over $n \times n$ matrices $$f:M \rightarrow M^T M$$. My teacher took for granted that it is continous. Intuituively, I'd say that $f$ is a polynomial with the entries ...
3
votes
1answer
88 views

Identity makes every matrix invertible?

I have found this in a proof and do not understand where this comes from: If A is singular, then there exists $\delta \in \mathbb{R}_{>0} \forall \epsilon\in (0,\delta): \epsilon ...
2
votes
1answer
95 views

Continuity of an $\mathbb {R}^2$ function

Let $f$ be an $\mathbb{R}^2$ endomorphism and $N:\mathbb{R}^2\to\mathbb{R}^+$ defined by $$\forall u \in \mathbb {R }^2, N(u) = ||f(u)|| $$ I need to show $N$ is continuous. The problem is that $N$ ...
5
votes
2answers
86 views

“Deformation” of the kernel of a linear map

It is known that the roots of a monic polynomial of fixed degree vary continuously (smoothly?) with its coefficients, at least over $\mathbb{C}$. My question is whether there is such a result for ...
3
votes
2answers
135 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
1
vote
0answers
57 views

Combined length of portions of a line segment

Suppose I have a continuous function $f : X \to \mathbb{R}^n$ (where $X \subseteq \mathbb{R}^n$) that is piecewise rigid, i.e. $X$ has a finite partition $\mathcal{P}$ such that for all $P \in ...
0
votes
1answer
113 views

Looking for not/continuous and differentiable function examples?

I have to give an example of functions which are: continuous and not differentiable $f(x)=|x|$ differentiable $f(x)=(1/2)* x *|x|$ not continuous $f(x)=1/x$ and $f(x)=1/cos(x) [0;\pi]$ Are ...
2
votes
1answer
74 views

Are the following linear maps continuous ?

I am supposed to find the whether the following maps are continuous or not , if continuous then to find the $||T||$ $P$ is a vector space of polynomials . Define norm on the polynomials $p\in P$ as ...
3
votes
2answers
221 views

How to show unitary decomposition is continuous

It is a well-known fact that, for $A \in GL(n,\mathbb{C})$ with polar decomposition $A=U_AP_A$ for $U_A$ unitary and $P_A$ positive definite and Hermitian, the map $GL(n,\mathbb{C}) \rightarrow U(n)$ ...