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Suppose $A \in M_n(\mathbb R)$ is an arbitrary square matrix and $\rho(A)$ is the spectral radius of $A$. Does this inequality hold: $$ \text{Trace}(A^{\top}A ) \ge \rho(A)?$$

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    $\begingroup$ If you scale $A$ by a factor $c>0$, the LHS is quadratic in $c$ while the RHS is linear in $c$. Therefore the inequality doesn't hold if the LHS is nonzero and $c$ is small. $\endgroup$ – user1551 Apr 23 at 8:11
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As @user1551 explained, a suitable counterexample could be $$ A = \begin{pmatrix} c & 0 \\ c & 0 \end{pmatrix} $$ for some $c \in (0,\frac{1}{2})$. Then, we have $$ \text{Tr}(A^T A) = \text{Tr} \begin{pmatrix} 2 c^2 & 0 \\ 0 & 0 \end{pmatrix} = 2c^2 < c = \rho(A). $$

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