Math 6610 • Analysis of Numerical Methods I

{Projectors}
Recall that a square matrix \(P\) that satisfies \(P^2=P\) is called a projector and that
the subspaces \(\operatorname{range}(P)\) and \(\operatorname{null}(P)\) are complementary
subspaces. Recall also that a projector \(P\) is an orthogonal projector if these two
subspaces are perpendicular to one another, or equivalently, if \(P\) is Hermitian.

\textbf{a)} Let \(P\) be a nonzero projector. Show that \(\lVert P\rVert_2\ge 1\), with
equality if and only if \(P\) is an orthogonal projector.

\textbf{b)} Given an \(m\times n\) matrix \(A\) \((m\ge n)\) with rank \(r\). Describe a
stable algorithm for constructing an orthogonal projector onto \(\operatorname{range}(A)\).
Why do you think your algorithm is stable?