Tag: choose

$newcommand{GLm}{text{GL}_n^-}$

Let $GLm$ be the space of real $n times n$ matrices with strictly negative determinant.

Define $X$ to be the set of all matrices $A in GLm$ whose smallest singular value of is strictly smaller than all the other singular values. $X$ is an open subset of $GLm$.

Can we choose locally the singular vectors of matrices in $X$ in a smooth manner?

Here is a more precise description of what I want:

Let $A in X$. Do there exist an open neighbourhood $O$ of $A$ in $X$, and smooth maps $U:O to text{SO}_n,V:O to text{O}_n^-$ such that
$$ A=U(A)Sigma(A)V(A)^T,$$ holds for every $A in O$, where
$Sigma(A) = text{diag}left( sigma_1(A),dotssigma_n(A) right)$, and $sigma_1(A)$ is the smallest singular value of $A$?

I don’t care about the ordering of $sigma_2,dots,sigma_n$, but I specifically want the minimal singular value to be in a fixed position.

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Comment 1: If we replace the requirement that $sigma_1 $ is strictly smaller than the others by the requirement that all the singular values are distinct, than the answer is positive. Indeed, in that case the map

begin{align*}
mu: text{SO}_ntimes mathcal Dtimes text{O}_n^-to X\
(U,Sigma,V)mapsto USigma V^T
end{align*}

is a local diffeomorphism, so is locally invertible (here $mathcal D$ is the space of $ntimes n$ diagonal matrices with strictly increasing positive entries).

Comment 2: If such maps $U,V$ exist, then the map $A to Sigma(A)$ is also smooth. In general the ordered singular values cannot be chosen smoothly when they “cross”, but here I don’t require to keep the ordering of $sigma_2,dots,sigma_n$ fixed, so I don’t think there is an obstruction, but I may be wrong.