I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is: $$ \Vert A \Vert_* = \sqrt{tr(M) + 2\sqrt{det(M)}}. $$

Are there any analytical form of nuclear norm for $n \times n$ matrix like the above form?

Or just for a $3 \times 3$ matrix?