Prove that the square sum of eigenvalues is no more than the Frobenius norm for a square complex matrix 
Let  $ \boldsymbol{A}=(a_{ij})_{n\times n} $ be a complex square matrix with eigenvalues: $\lambda_1 ,\lambda_2, \dots , \lambda_n $.  Prove that
$$ \sum_{r=1}^{n} |\lambda_r|^2 \le \sum_{i,j=1}^{n} |a_{ij}|^2\,.$$
Furthermore, show that the equality holds if and only if $$\boldsymbol{A^H A=AA^H}\,.$$
Here, $\boldsymbol{A^H}$ is the conjugate transpose of $\boldsymbol{A}$.

We can write $$\sum_{i,j=1}^n\,|a_{ij}|^2=\operatorname{Tr}(\boldsymbol{AA^H})=\|\boldsymbol{A}\|_F^2,$$
where $\|\_\|_F$ is the Frobenius norm.  So this inequality gives a lower bound on $\|\boldsymbol{A}\|_F$, namely
$$\|\boldsymbol{A}\|_F\geq \sqrt{\sum_{r=1}^n\,|\lambda_r|^2}\,.$$
Edit.  Another proof of the inequality is found here.  However, there is no discussion about the equality case.
 A: If $\mathbf{A}$ is a general square complex matrix (i.e. $\mathbf{A}\in\mathbb{C}^{n\times n}$), we have by Schur decomposition: 
$$\mathbf{A}=\mathbf{Q}\mathbf{U}\mathbf{Q}^{\dagger}$$
Where $\mathbf{Q}$ is a unitary matrix (i.e. $\mathbf{Q}^{-1}=\mathbf{Q}^{\dagger}$, where $\mathbf{Q}^{\dagger}$ represents the conjugate transpose of $\mathbf{Q}$) and $\mathbf{U}$ is an upper triangular matrix. Multiplying each side by its Hermitian transpose, we get:
$$\mathbf{A}\mathbf{A}^{\dagger}=(\mathbf{Q}\mathbf{U}\mathbf{Q}^{\dagger})(\mathbf{Q}\mathbf{U}\mathbf{Q}^{\dagger})^{\dagger}=\mathbf{Q}\mathbf{U}\mathbf{Q}^{\dagger}\mathbf{Q}\mathbf{U}^{\dagger}\mathbf{Q}^{\dagger}=\mathbf{Q}\mathbf{U}\mathbf{U}^{\dagger}\mathbf{Q}^{\dagger}$$
Taking the trace of both sides and using the property that the Frobenius norm of $\mathbf{A}$ is $\operatorname{Tr}(\mathbf{A}\mathbf{A}^{\dagger})$, we have:
$$\sum_{1\leq i,j \leq n}|A_{ij}|^{2}=\operatorname{Tr}(\mathbf{A}\mathbf{A}^{\dagger})=\operatorname{Tr}(\mathbf{Q}\mathbf{U}\mathbf{U}^{\dagger}\mathbf{Q}^{\dagger})=\operatorname{Tr}(\mathbf{U}\mathbf{U}^{\dagger})=\operatorname{Tr}((\mathbf{\Lambda}+\mathbf{N})(\mathbf{\Lambda}+\mathbf{N})^{\dagger})$$
We get the final equality from the property of an upper triangular matrix that it can be written as the sum of a strictly upper triangular matrix $\mathbf{N}$ and a diagonal matrix, which in this case is the matrix with eigenvalues along its diagonal $\mathbf{\Lambda}=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$. We also use the property that the conjugate transpose is distributive over matrix addition, i.e. $(\mathbf{\Lambda}+\mathbf{N})^{\dagger}=\mathbf{\Lambda}^{\dagger}+\mathbf{N}^{\dagger}$. Therefore we have:
$$\sum_{1\leq i,j \leq n}|A_{ij}|^{2}=\operatorname{Tr}((\mathbf{\Lambda}+\mathbf{N})(\mathbf{\Lambda}^{\dagger}+\mathbf{N}^{\dagger}))=\operatorname{Tr}(\mathbf{\Lambda}\mathbf{\Lambda}^{\dagger}+\mathcal{O}(\mathbf{\Lambda}))$$
And therefore because $\operatorname{Tr}(\mathbf{\Lambda}\mathbf{\Lambda}^{\dagger}) \leq \operatorname{Tr}(\mathbf{\Lambda}\mathbf{\Lambda}^{\dagger} + \mathcal{O}(\mathbf{\Lambda}))$, we have: 
$$\sum_{i=1}^{n}|\lambda_{i}|^{2}\leq \sum_{1\leq i,j \leq n}|A_{ij}|^{2}$$
As required.
Note that if a matrix $\mathbf{A}$ is normal, i.e. it satisfies $\mathbf{A}^{\dagger}\mathbf{A}=\mathbf{A}\mathbf{A}^{\dagger}$  then $\mathbf{A}$ is diagonalizable, i.e. $\exists \mathbf{P}\in\mathbb{C}^{n\times n}$ such that $\mathbf{A}=\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{\dagger}$ and $\mathbf{\Lambda}=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$ and $\mathbf{P}$ is unitary. 
Multiplying by the complex conjugate of both sides:
$$\mathbf{A}\mathbf{A}^{\dagger}=(\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{\dagger})(\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{\dagger})^{\dagger}=\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{\dagger}\mathbf{P}\mathbf{\Lambda}^{\dagger}\mathbf{P}^{\dagger}=\mathbf{P}\mathbf{\Lambda}\mathbf{\Lambda}^{\dagger}\mathbf{P}^{\dagger}$$
Again using the property that the Frobenius norm is $\operatorname{Tr}(\mathbf{A}\mathbf{A}^{\dagger})$, we take the trace of both sides:
$$\sum_{1 \leq i,j \leq n}|A_{ij}|^{2}=\operatorname{Tr}(\mathbf{A}\mathbf{A}^{\dagger})=\operatorname{Tr}(\mathbf{P}\mathbf{\Lambda}\mathbf{\Lambda}^{\dagger}\mathbf{P}^{\dagger})=\operatorname{Tr}(\mathbf{\Lambda}\mathbf{\Lambda}^{\dagger})=\sum_{i=1}^{n}\lambda_{i}\lambda_{i}^{*}=\sum_{i=1}^{n}|\lambda_{i}|^{2}$$ 
So we have equality iff $\mathbf{A}$ is normal.
