The Schatten p-norm of a matrix is given by

$$\|A\|_p = \left[\text{tr}\left((A^*A)^{p/2}\right)\right]^{1/p}$$

Some of them are induced vector norms. For example, $\|A\|_\infty = \max\{\|Au\|: \|u\|= 1\}$ is the operator norm and $\|\cdot\|$ is the Euclidean 2-norm for vectors.

Can all the other Schatten p-norms be written as induced vector norms? More generally, is there any connection between the vector p-norms and the matrix Schatten p-norms?


1 Answer 1


The Schatten. $p$-norm of a matrix $A\in M_n(\mathbb C)$ is the $p$-norm of the vector containing the singular values of $A$. That is, $\|A\|_p=\|\left(\sigma_1(A),\ldots,\sigma_n(A)\right)\|_p$. When $A$ is a square matrix whose size is at least $2\times2$, $\|A\|_p$ is not an induced norm unless $p=\infty$, because $\|I_n\|_p=n^{1/p}>1$ when $p$ is finite.


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