# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $M \rightarrow M^T M$ 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 ...
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### 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 ...
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 ...
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 $GL(n,\mathbb{C}) \rightarrow U(n)$ ...