Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ (with respect to the inner product $(A,B) \mapsto \operatorname{tr}(A^\top B)$) and we denote by $r$ its dimension.
What relationship holds between $d$ and $r$?