TITLE: Do these matrices have a name?
QUESTION [2 upvotes]: I'm wondering if these matrices have a name? (I'm somehow tempted to call them subunitary but it seems to be reserved for something else.) 
The matrix $M \in \mathbb{C}^{n\times n}$ is called ..., if  all the singular values $\sigma_1,\dots, \sigma_n$ are strictly smaller than 1. 
Note that if all the singular values are 1 then $M$ is a unitary matrix. That is why I think it should be called subunitary.

The matrix $M \in \mathbb{C}^{n\times n}$ is called ..., if  it fulfills
$$M = \alpha U$$
with $|\alpha|<1, \alpha \in \mathbb{C}$ and $U \in U(n)$ ($U(n)$ is the group of unitary matrices). (maybe it has a name if one allows $|\alpha|=1$)
It seems like this space is closed under multiplication. But obviously it is not a group as the $0$ matrix is not invertible.

REPLY [2 votes]: If you allow the eigenvalues to have different norms, all at most $1$, then it's a contraction.
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