The following result gives a lower bound for the rank of a square matrix in terms of its trace and Frobenius norm (a.k.a. Hilbert-Schmidt norm).

Lemma 1Let be any matrix with complex entries. ThenEquality 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 .

This weakening is usually of no concern in applications. However, in this paper of mine I needed the sharp estimate (1) for general non-symmetric (although real) matrices.

Before giving a proof of Lemma 1, I present the following application, which is a slight strengthening of Lemma 2 of this post by Terence Tao.

Lemma 2Let . 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,

and (1) follows from (2), (3), (4) and (5).

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.

Conversely, if is normal, then is diagonal, and equality holds in (2) and (4). If all the non-zero eigenvalues of are equal, equality holds in (3) and (5), and we obtain equality in (1).