[Math] Trace and Spectral norm of a matrix

Question

Let $A_{n\times n}$ be a matrix. How I can show
$$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$
and if $A$ is symmetric and positive definite,
$$\operatorname{trace}(A) \geqslant \Vert A \Vert_2$$

Answer 1

What is the spectral radius of $A^TA$? it is its largest eigenvalue. The square root of the largest eigenvalue of $A^TA$ is the largest singular value. Since the trace is the sum of eigenvalues, the inequality follows. If $A$ is symmetric and real, all eigenvalues are real, and if it is positive definite, all eigenvalues are positive. Since the spectral norm is the largest eigenvalue, the result follows.