Projection matrices in linear algebra: an introduction (orthogonal, oblique, pseudoinverse)

This video outlines the concepts of orthogonal and oblique projections onto a subspace, with a graphical 2D/3D example and later on generalising the ideas to an arbitrary vector space (finite-dimensional), where a linear transformation can be represented by a matrix P.

A matrix P is a projection matrix if P^2=P. Projection is orthogonal if P^T(I-P)=0 and that is equivalent to P being a symmetric projection matrix. Also, it is shown that projection matrix eigenvalues can be either zero or one.
A couple of examples illustrate the concepts (particularly, the pseudo-inverse one, cornerstone of least-squares techniques).
Oblique projection matrix to column space of A in direction B is also presented (without proof).

#projection #matrix #geometry #linearalgebra
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Antonio Sala
Universitat Politecnica de Valencia, Spain