Hi All,


Let us consider a P x Q real matrix (P >= Q). It can be thought of as an element of $\mathbb{R}^{PQ}$. We are considering Lebesgue measure over that space. My question is whether the subspace of matrices with repeated singular values are of measure 0 or not. 


Any suggestions would be welcome.


Thanks
Ashin



Measure of Subspace of Matrices with repeated Singular Values
The eigenvalues of
$$
 \widehat M = \begin{pmatrix}0&M\\ M^T&0\end{pmatrix}
$$
are the squares of the singular values of $M$.
A symmetric $n\times n$ matrix $N$ has a repeated eigenvalue if and only if the rank of
$$
 N\otimes I - I\otimes N
$$ 
is less than $n^2-n$. So the set of matrices $\widehat M$ with a repeated eigenvalue is
a proper subvariety of the set of matrices $\widehat M$, hence this set will have
measure zero.


