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

Lemma. Let $\text{[math]}$ , and let $\text{[math]}$ where $\text{[math]}$ is the space of row-null column-null $\text{[math]}$ matrices. Then $\text{[math]}$ if and only if $\text{[math]}$ 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 $\text{[math]}$ equations, only $\text{[math]}$ of which are linearly independent.

Consider the following space

Its dimension is

where $\text{[math]}$ is the contribution from the antisymmetric part and $\text{[math]}$ is from the symmetric part.

Assume $\text{[math]}$ and $\text{[math]}$ , then the Frobenius inner product of two such elements is

Since $\text{[math]}$ and $\text{[math]}$ , then $\text{[math]}$ and $\text{[math]}$ must be complementary in $\text{[math]}$ . Therefore, if $\text{[math]}$ is orthogonal to all the matrices in $\text{[math]}$ , it must lie in $\text{[math]}$ .