An inequality about norms of matrices

Asked Mar 28 '19 at 11:53

Active Mar 28 '19 at 11:53

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I encountered an inequality that I have no idea how to prove.

$ |\mathrm{Tr}(A^*B)| \leq ||A|| \cdot ||B||_1 $, where $A^*$ is the adjoint of $A$, and $||A||$ denotes the spectral (or operator) norm of $A$, and $||B||_1$ denotes the trace norm of $B$.

Maybe the first step we can do is to assume w.l.o.g. B is diagonal. But what to do then?

asked Mar 28 '19 at 11:53

Lwins

You are doing this for matrices? For completeness, please define the operator and trace norm as you know them (the definition you have). – Sarvesh Ravichandran Iyer Mar 28 '19 at 12:03
2

You may want to see this: https://math.stackexchange.com/questions/2241879/reference-for-trace-norm-inequality. – Minus One-Twelfth Mar 28 '19 at 12:03

An inequality about norms of matrices

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