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(\beginarrayccAB\\B^*D\endarray\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.