TABLE OF CONTENTS We now enter the second part of the book, whose theme is orthogonality. We begin with the fundamental tool of projection matrices, or projectors, both orthogonal and nonorthogonal.
A projector is a square matrix P that satisfies
(Such a matrix is also said to be idempotent.) This definition includes both orthogonal projectors, to be discussed in a moment, and nonorthogonal ones. To avoid confusion one may use the term oblique projector in the nonorthogonal case.
The term projector might be thought of as arising from the notion that if one were to shine a light onto the subspace range ( P) from just the right direction, then Pv would be the shadow projected by the vector v. We shall carry this physical picture forward for a moment.
Observe that if v ? range ( P), then it lies exactly on its own shadow, and applying the projector results in v itself. Mathematically, we have v = Px for some x and
From what direction does the light shine when v ? Pv? In general the answer depends on v, but for any particular v, it is easily deduced by drawing the line from v to Pv, Pv ? v (Figure 6.1). Applying the projector to this...
