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2 votes

Generalisation of Jordan decomposition to rectangular matrices

(The answer is completely replaced) Consider together with $M$ another $m \times n$ matrix $N = [I|0]$. Transformations $M \mapsto AMT^{-1}$ with $T = \begin{bmatrix}A & 0 \\ B & C\end{bmatrix}...
Vladimir Lysikov's user avatar
4 votes

Continuous path of unitary matrices with prescribed first column?

Question 1: yes As pointed out in the comments by @მამუკაჯიბლაძე, the map $U(n)\to S^{2n-1}$ taking a unitary matrix to its first column is a fiber bundle and therefore a fibration. Question 2: no Let ...
Tom Goodwillie's user avatar
2 votes

Does this matrix equation always have a solution?

No. Here is an explicit counter-example for the $i = 3$ case: $$ A_3^\prime = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 &...
Nathaniel Johnston's user avatar
1 vote

Expected value of the largest singular value of a random matrix with entries in $N (0,1)$

Largest eigenvalue $x$ of $A'A/n$ converges to scaled/shifted Tracy-Widom distribution. More specifically, the following quantity follows Tracy-Widom distribution. $$\frac{(x-4) n^{2/3}}{2 \sqrt[3]{2}}...
Yaroslav Bulatov's user avatar
2 votes

Wold decomposition of toral endomorphisms

The Hilbert space $H := L^2(\mathbb{T}^d,dx)$ can be canonically identified with the subspace of all (classes of) $\mathbb{Z}^d$-periodic locally square-integrable functions on $\mathbb{R}^d$. Then, ...
Christophe Leuridan's user avatar
2 votes

Calculate the Riemannian Hessian of Karcher mean problem on positive definite matrices

I came up with an ad-hoc solution to calculate the Hessian-vector product below. I'm still looking forward to see if there are any comments on the more general question, i.e. calculating the ...
Jason Li's user avatar
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