Projection matrix
Projection matrix
We wish to project the vector \( b \) onto the vector \( a \). The projection matrix \( P \) achieves this by \( Pb \), and is given by:
\[ P = \frac{aa^T}{a^Ta} \]We expect that a projection matrix \( P \):
- Is singular.
- Has columns that are multiples of \( a \).
- Is symmetric.
- Has eigenvalues 1 and 0 corresponding to vectors \( a \) and vectors perpendicular to \( a \).
Dot product approach
The intuitive projection is to normalize \( a \) to be a unit vector then take the dot product with \( b \) to produce a scalar, and then scale the normalized \( a \) vector by this scalar.
Matrix approach
The matrix approach preloads \( a \) into the columns of a matrix \( P \), and scales the columns by and amount that both:
- normalizes \( a \)
- partially-computes the dot product
Matrix multiplication \( Pb \) then amounts to \( b_1 \) scaling the normalized \( a \) by the value \( b_1a_1 \), and so on for \( b_2, b_3 \) etc., and summing these intermediate vectors.
Compute
It's interesting that the intuitive approach is cheaper than the matrix approach.
