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

**Lemma.** Let , and let
where is the space
of row-null column-null matrices. Then
if and only if 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 equations, only of which are linearly independent.

Consider the following space

Its dimension is

where is the contribution from the antisymmetric part and is from the symmetric part.

Assume and , then the Frobenius inner product of two such elements is

Since and , then and must be complementary in . Therefore, if is orthogonal to all the matrices in , it must lie in .