Projections

When a vector does not exist in a column space, the projection is the best approximation of it in linear combinations of that column space.

Vectors

Given vectors a and b, a can be projected into C(b), the column space of b. This projection p has an error term e.

Trigonometric Approach

Projections with vectors can be calculated in terms of θ is the angle formed by a and b.

A vector in the direction of b with the magnitude of a is given by ||b|| cos(θ). This can be called the scalar projection.

However, a vector projection should have a magnitude based on how much a moved through C(b). This is captured by â, the unit vector in the direction of a, which can be calculated as a/||a||. The projection vector is given by (||a|| cos(θ)) (a/||a||) = (||b|| cos(θ)) â.

Algebraic Approach

Projections with vectors can also be calculated in terms of the vectors themselves, as they represent linear transformations.

First, the dot product can be substituted into the above formulas to give a scalar projection as a⋅b/||a|| and a vector projection as (a⋅b/||a||) a/||a|| = (a⋅b/||a||) â.

The vector projection can then be reformulated like:

p = (a⋅b/||a||) a/||a||

p = (a⋅b/||a||²) a

p = (a⋅b/a⋅a) a

or:

p = (a⋅b/||a||) â

p = (â⋅b) â

Linear Algebraic Approach

The linear transformation from vector a to projection vector p is expressed as p = ax̂. The projection carries an error term that can be characterized by e = b - p or e = b - ax̂. a is orthogonal to e, so a⋅(b - ax̂) = 0. This simplifies to x̂ = (a⋅b)/(a⋅a). Altogether, the projection vector is p = a (a⋅b)/(a⋅a).

The projection matrix P satisfies p = Pb. C(P), the column space of P, is equivalent to C(a). It follows that P is also of rank 1.

Properties

The projection matrix P is symmetric (i.e. P^T = P) and idempotent (i.e. P² = P).

Matrices

Given a system as Ax = b, if b is not in C(A), the column space of A, then there is no possible solution for x. The best approximation is expressed as Ax̂ = p where projection p estimates b with an error term e.

The error term can be characterized by e = b - p or e = b - Ax̂. e is orthogonal to R(A), the row space of A; equivalently it is orthogonal to C(A^T). Orthogonality in this context means that e is in the null space, so A^T(b - Ax̂) = 0.

The system of normal equations is A^TAx̂ = A^Tb. This simplifies to x̂ = (A^TA)^-1A^Tb. Altogether, the projection is characterized by p = A(A^TA)^-1A^Tb.

The projection matrix P satisfies p = Pb. It is calculated as P = A(A^TA)^-1A^T.

b can also be projected onto e, which geometrically means projecting into the null space of A^T. Algebraically, if one projection matrix has been computed as P, then the projection matrix for going the other way is (I - P)b.

Properties

As above, the projection matrix P is symmetric and idempotent.

If A is square, the above equations simplify rapidly.

If b actually was in C(A), then P = I. Conversely, if b is orthogonal to C(A), then Pb = 0 and b = e.

Usage

This should look familiar. A projection is inherently the minimization of the error term.

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