Submatrix of Orthogonal Projection

Let $\mathcal {H}$ be a n-dimensional complex Hilbert space, and $Q$ be an orthogonal projection on $\mathcal {H}.$ If $\mathcal {M}$ is a subspace of $\mathcal {H}$ and $\dim{\mathcal {M}}=r,$ then under the space decomposition $\mathcal {H}=\mathcal {M}\oplus\mathcal {M}^{\perp}$, $Q=\left(\begin{array}{cc}AB\\B^*D\end{array}\right),$where $A\in{\mathcal {B}}({\mathcal {M}}), B\in{\mathcal {B}}({\mathcal {M}}^{\perp},{\mathcal {M}}), D\in\mathcal {B}(\mathcal {M}^{\perp}).$ In this paper, using of the technique of block operator matrix, the properties and relations between $A, B$ and $D$ are given. Furtherly, the relations between $P$ and $Q$ are discussed, where $P$ is an orthogonal projection on $\mathcal {M},$ (i) ${\mathcal {R}}(P)\cap{\mathcal {R}}(Q)=$\{0\}$\Leftrightarrow \dim {\mathcal {R}}(A)=\dim {\mathcal {R}}(B),$ (ii) ${\mathcal {R}}(P)+{\mathcal {R}}(Q)={\mathcal {H}} \Leftrightarrow \dim {\mathcal {R}}(D)=n-r,$ (iii) ${\mathcal {R}}(P)\perp{\mathcal {R}}(Q) \Leftrightarrow \dim {\mathcal {R}}(A)=0$ are obtained.