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Questions about the properties of vector spaces and linear transformations, including linear systems in general.

5 votes
Accepted

If $A^TA \ge B^TB$ does this imply $AA^T \ge BB^T$?

Take $$A = \begin{pmatrix}0 & 0 \\ 1 & 0\end{pmatrix}, \quad B = \begin{pmatrix}1 & 0 \\ 0& 0 \end{pmatrix}.$$
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2 votes
0 answers
751 views

Reference: Continuity of Eigenvectors [closed]

I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, but got no answer. For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix …
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1 vote
Accepted

Is this notion of being "fully" convex closed under set addition?

Have a look at the example from https://mathoverflow.net/a/37683/32507. This gives two convex, closed sets which cannot be separated by linear functionals (even not by discontinuous ones). If I unders …
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