Projection Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


As discussed in §5.9.9, the orthogonal projection of $ y\in{\bf C}^N$ onto $ x\in{\bf C}^N$ is defined by

$\displaystyle {\bf P}_{x}(y) \isdef \frac{\left<y,x\right>}{\Vert x\Vert^2} x.

In matlab, the projection of the length-N column-vector y onto the length-N column-vector x may therefore be computed as follows:
yx = (x' * y) * (x' * x)^(-1) * x
More generally, a length-N column-vector y can be projected onto the $ M$-dimensional subspace spanned by the columns of the N $ \times$ M matrix X:
yX = X * (X' * X)^(-1) * X' * y
Orthogonal projection, like any finite-dimensional linear operator, can be represented by a matrix. In this case, the $ N\times N$ matrix
PX = X * (X' * X)^(-1) * X'
is called the projection matrix.I.2Subspace projection is an example in which the power of matrix linear algebra notation is evident.

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]