Equality holds in (1) if and only if is a normal matrix and all its non-zero eigenvalues are equal. If is a real matrix then equality holds in (1) if and only if is symmetric and all its non-zero eigenvalues are equal.
The special case where is real and symmetric is an exercise on p.~137 of Bellman’s of a square matrix, and looking at Schur’s original 1909 paper gives the impression that Schur could have known this.
Various combinatorial and geometric applications have been given by Noga Alon and others (see my paper for more references). These papers all use (1) only for symmetric matrices. If is not symmetric, the following argument implies a weakening of (1) by a factor of .
Apply (1) to the symmetric matrix , which has rank at most , to obtain
where the first inequality follows from the Cauchy-Schwarz inequality in the form .
Lemma 2 Let . Let be unit vectors in such that for all distinct . Then .
In particular, if for all distinct , then .
Proof: Consider the Gram matrix of . Its rank is , its trace is , and the square of its Frobenius norm satisfies . Substitute back into (1) and solve for to obtain
We end with the proof of Lemma 1.
Proof: Let the non-zero eigenvalues of be . Since the result is trivial if , we may assume without loss of generality that . By the Schur decomposition of a square matrix with complex entries there exists an unitary matrix such that is upper triangular. In particular, the eigenvalues of are the diagonal entries of , and
(This inequality is in Schur’s 1909 paper.) Finally, by the Cauchy-Schwarz inequality,
Suppose that equality holds in (1). This gives equality in (2)—(5). Equality in (5) gives that all are equal. Equality in (3) implies that all are positive multiples of each other. Therefore, all are equal. Equality in (4) gives that is a diagonal matrix, hence is normal. If is real we furthermore obtain that the are real, since they are equal and their sum is the real number . Then , hence and is symmetric.