Projection
As discussed in §5.9.9,
the orthogonal projection of
onto
is defined by
![$\displaystyle {\bf P}_{x}(y) \isdef \frac{\left<y,x\right>}{\Vert x\Vert^2} x.
$](http://www.dsprelated.com/josimages_new/mdft/img2108.png)
yx = (x' * y) * (x' * x)^(-1) * xMore generally, a length-N column-vector y can be projected onto the
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
![$ \times$](http://www.dsprelated.com/josimages_new/mdft/img1735.png)
yX = X * (X' * X)^(-1) * X' * yOrthogonal projection, like any finite-dimensional linear operator, can be represented by a matrix. In this case, the
![$ N\times N$](http://www.dsprelated.com/josimages_new/mdft/img2092.png)
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.
Projection Example 1
>> X = [[1;2;3],[1;0;1]] X = 1 1 2 0 3 1 >> PX = X * (X' * X)^(-1) * X' PX = 0.66667 -0.33333 0.33333 -0.33333 0.66667 0.33333 0.33333 0.33333 0.66667 >> y = [2;4;6] y = 2 4 6 >> yX = PX * y yX = 2.0000 4.0000 6.0000
Since y in this example already lies in the column-space of X, orthogonal projection onto that space has no effect.
Projection Example 2
Let X and PX be defined as Example 1, but now let
>> y = [1;-1;1] y = 1 -1 1 >> yX = PX * y yX = 1.33333 -0.66667 0.66667 >> yX' * (y-yX) ans = -7.0316e-16 >> eps ans = 2.2204e-16
In the last step above, we verified that the projection yX is
orthogonal to the ``projection error'' y-yX, at least to
machine precision. The eps variable holds ``machine
epsilon'' which is the numerical distance
between and the next representable number in double-precision
floating point.
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Orthogonal Basis Computation
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Vector Cosine