Frobenius norm and eigenvalue sum of squares via Schur decomposition

Expert intro: The key idea is that unitary similarity preserves the Frobenius norm, while the upper triangular Schur form makes the diagonal visible. That diagonal is exactly where the eigenvalues sit.

Detailed walkthrough

Let A be an n\times n complex matrix. By Schur decomposition, there exists a unitary matrix U and an upper triangular matrix T such that $A=UTU^*,$ and the diagonal entries of T are the eigenvalues of A: $T_{11}=\lambda_1,\dots,T_{nn}=\lambda_n.$

Now use unitary invariance of the Frobenius norm: $\|A\|_F^2=\operatorname{tr}(A^*A)=\operatorname{tr}((UTU^*)^*(UTU^*))=\operatorname{tr}(UT^*T U^*)=\operatorname{tr}(T^*T).$ So $\|A\|_F^2=\sum_{i,j}|T_{ij}|^2.$ Because T is upper triangular, this sum contains all diagonal terms and all off-diagonal terms above the diagonal: $\|A\|_F^2=\sum_{i=1}^n |T_{ii}|^2 + \sum_{i<j}|T_{ij}|^2.$ Since the diagonal entries of T are the eigenvalues of A, $\sum_{i=1}^n |\lambda_i|^2 = \sum_{i=1}^n |T_{ii}|^2 \le \sum_{i=1}^n |T_{ii}|^2 + \sum_{i<j}|T_{ij}|^2 = \|A\|_F^2.$ This proves $\sum_{i=1}^n |\lambda_i|^2\le \|A\|_F^2.$

Equality holds exactly when every off-diagonal entry of T is zero, i.e. when T is diagonal. In that case A is unitarily diagonalizable, equivalently A is normal: $AA^*=A^*A.$

💡 Pitfall guide

The inequality is not coming from eigenvalues alone; it comes from splitting the Frobenius norm into diagonal and off-diagonal parts in Schur form. If you skip the unitary invariance step, the argument looks incomplete. Also, equality does not hold for every diagonalizable matrix, only for those that are unitarily diagonalizable.

🔄 Real-world variant

If A happens to be real symmetric or complex Hermitian, then Schur form is actually diagonal, so equality always holds and the inequality becomes $\sum_{i=1}^n \lambda_i^2=\|A\|_F^2$ when the eigenvalues are real. For a general diagonalizable matrix, the off-diagonal terms in a non-unitary basis can still be present, so diagonalizability alone is not enough.

🔍 Related terms

Schur decomposition, Frobenius norm, unitary diagonalization