# the distribution of Singular value of rectangular gaussian matrix

the singular value decomposition of an $$m\times n$$ random Gaussian matrix $${\displaystyle \mathbf {M} }$$ is a factorization of the form $${\displaystyle \mathbf {U\Sigma V^\ast} }$$, $${\displaystyle \mathbf {\Sigma } }$$ is an $${\displaystyle m\times n}$$ rectangular diagonal matrix with non-negative ordred real numbers on the diagonal, my question is:

What is the distribution of the singular values of $$\Sigma$$?

can I say that the singular value corresponds to the absolute value of Gaussian variable?

With fixed ratio $$\lambda=m/n$$ the Marchenko-Pastur distribution gives the asymptotic distribution of singular values for a Gaussian rectangular matrix.
• and what can I say if $m$ is fixed and $n$ is arbitrarily large? Nov 14, 2022 at 8:13