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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.

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    @Jonas Meyer: I accepted the answer, because it actually corresponds to the answer which I was searching for. In the beginning, I thought my matrices are of the form $\alpha U$. Closer inspection did show, that the eigenvalues in my application do in fact have different norms (in the mean time I figured out that my matrices might not be invertible at all). I will edit the question with the exact definition of the matrices whose name I'm searching for. I will inform Yuval of the change.2011-05-17

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If you allow the eigenvalues to have different norms, all at most $1$, then it's a contraction.

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    @Fabian: Thank you also, Fabian, for clarifying. It was no trouble.2011-05-17