Projection Matrices: quick introduction

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 15:13

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Abstract/Summary:

This video outlines the concepts of orthogonal and oblique projections, 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 $P$.

Such matrix $P$ is a projection matrix if $P^2=P$. Projection is orthogonal if $P^T(I-P)=0$ amd 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).