Characterization of the oblique projector \(U(VU)^{\dagger}V\) with application to constrained least squares (Q840653)

The author provides a full characterization of the oblique projector \(U(VU)^{\dagger}V\) with application to constrained least squares. For any two complementary subspaces \(L\) and \(M\) of \(\mathbb C^m\), the oblique projector onto \(L\) along \(M\) is denoted by \(P_{L,M}\). The author proves that if \(L\) and \(M\) are complementary subspaces of \(\mathbb C^m\), for any two matrices \(U,V\), with \(R(U)=L\) and \(N(V)=M\), one has \[ P_{L,M}=U(VU)^{\dagger}V, \] where \(A^{\dagger}\) denotes the Moore-Penrose inverse of \(A\). In the general case, where the range of \(U \in \mathbb C^{m \times p}\) and the null space of \(V \in \mathbb C^{q \times m}\) are not complementary subspaces, the author shows that matrix \(E=U(VU)^{\dagger}V\) is idempotent with range and null space given by \[ \begin{aligned} R(E)&=R(UU^*V^*)=R(UU^*V^*V)=R(U) \cap \left( (UU^*)^{\dagger} (R(U) \cap N(V)) \right)^{\bot}, \\ N(E)&=N(U^*V^*V)=N(UU^*V^*V)=N(V) \oplus \left( (V^*V)^{\dagger} (R(U)+N(V)) \right)^{\bot}, \end{aligned} \] where \(A^*\) denotes the conjugate transpose of matrix \(A\). Finally, the author analyzes the last result in the context of constrained least squares minimization.