# The Orthogonal Complement of the Space of Row-null and Column-null Matrices

**Lemma.** Let \(Z\in\text{GL}(n,R)\), and let
\(Y\in\mathcal{S}(n,R)\) where \(\mathcal{S}(n,R)\) is the space
of row-null column-null \(n\times n\) matrices. Then
\(\text{Tr}(ZY)=0\) if and only if \(Z\) has the form

**Proof.** Consider the space of row-null and column-null matrices

Its dimension is

since the row-nullness and column-nullness are defined by \(2N\) equations, only \(2N-1\) of which are linearly independent.

Consider the following space

Its dimension is

where \(N-1\) is the contribution from the antisymmetric part and \(N\) is from the symmetric part.

Assume \(Y\in\mathcal{S}\) and \(Z\in\mathcal{G}\), then the Frobenius inner product of two such elements is

Since \(\text{dim}(\mathcal{G})+\text{dim}(\mathcal{S})=\text{dim}(GL)\) and \(\mathcal{G}\perp\mathcal{S}\), then \(\mathcal{G}\) and \(\mathcal{S}\) must be complementary in \(GL\). Therefore, if \(Y\) is orthogonal to all the matrices in \(\mathcal{S}\), it must lie in \(\mathcal{G}\).