# (p.203) Appendix C Orthogonal projections

# (p.203) Appendix C Orthogonal projections

The orthogonal projection **B** of a point **A** onto a linear space *L* is such that the direction **AB** is orthogonal to all directions in the space; it follows that **B** is the nearest point in *L* to A. Figure C.1 illustrates the geometry:

If **A** _{1} has coordinates **x** and **B** _{1} has coordinates **y** and the columns of _{P} **L** _{R} are independent and span the **R**-dimensional subspace *L*, then because **A** _{1} **B** _{1} is orthogonal to *L*, (**x** − **y**)**L** = 0 and because **y** ∈ **L**, there exists an *R*-dimensional column-vector **c** such that **y′** = **Lc**. It follows that:

**L**to be orthonormal, so that

**L′L**=

**I**, we have the simplified form:

**B**relative to the original

*P*-dimensional axes. Relative to the

*R*orthonormal axes in

*L*, (C.2) becomes;

*P*dimensions or a coordinate system embedded in the

*R*-dimensional subspace.

When the columns of **L** are orthonormal we may regard them as the first *R* columns of some *P*-dimensional orthogonal matrix whose final *P*−*R* columns are irrelevant, so long as they are normal to each other and to the first *R* columns. Thus, if we partition an orthogonal matrix **Q** = (**Q** _{1}, **Q** _{2}), then we have:

We note that we must have *R* ≤ *P* _{p}. We have seen that *R* = *P* gives an orthogonal matrix. It is impossible to have **L′L** = **I** when *R* > *P* because, geometrically, it would require the existence of *R* orthogonal directions in a space of fewer than *R* dimensions.

Returning to the general form (C.1) we shall write:

**P**

^{2}=

**P**, so that the matrix is idempotent. This means that once A is projected to B, repeated projection will leave the point B unchanged, which is obvious geometrically.

# C.1 Orthonormal operators as rotations

Above, **xL** was interpreted as **a** projection from a *P*-dimensional space onto a subspace spanned by the *Q* orthogonal columns of **L**. What might be the interpretation of **xL′**? Because *Q* < *P*, this certainly means that a point in *q* dimensions is transformed into a point in a higher dimensional space of *P* dimensions. The columns of **L′** are not orthonormal, so **xL′** does not represent projection. We have that (**xL′**)(**xL′**)′ = **xx′**, so inner products are invariant to the transformation and, hence, so are distances. Unlike with projections, **xL′** preserves configurations and hence it must represent a generalised rotation of **x** into the higher dimensional space. Another way of seeing this is to write:

**x**is padded out by

*P*−

*Q*extra zero coordinates and

**L**┴ represents a set of orthogonal columns in the orthogonal complement of

**L**. Now (

**L, L**

_{┴}) is an orthogonal matrix of order

**and so represents an orthogonal rotation in**

*P**P*dimensions.