### Projection

As discussed in §5.9.9, the orthogonal projection of onto is defined by

`N`column-vector

`y`onto the length-

`N`column-vector

`x`may therefore be computed as follows:

yx = (x' * y) * (x' * x)^(-1) * xMore generally, a length-

`N`column-vector

`y`can be projected onto the -dimensional

*subspace*spanned by the columns of the

`N`

`M`matrix

`X`:

yX = X * (X' * X)^(-1) * X' * yOrthogonal projection, like any finite-dimensional

*linear*operator, can be represented by a matrix. In this case, the matrix

PX = X * (X' * X)^(-1) * X'is called the

*projection matrix*.

^{I.2}Subspace 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.

**Next Section:**

Orthogonal Basis Computation

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Vector Cosine