For $A,B\in\mathbb{R}^{n\times m}$ we have the trace duality property $$|\langle A, B \rangle|\leq \|A\|_1 \|B\|_{\infty}$$ where $\|A\|_p$ is the Schatten $p$-norm (i.e. $\|\cdot \|_1$ is the nuclear norm equal to the sum of singular values, and $\|\cdot\|_{\infty}$ is the operator norm equal to the largest singular value) and the inner product is $\langle A, B \rangle = \text{tr}(A^{\top}B)$.
There are at least two methods to prove this inequality. One is using the Fischer-Courant min-max principle (see for example this question), and the other is by the aid of symmetric gauge functions (see Chapter 4 of Matrix Analysis (1997) form Bathia). None of these proofs establish sufficient (or necessary) conditions to get an equality.
Do anyone know a way to get equality?
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: $|$. – joriki Mar 29 '20 at 19:37