The ProjectionMatrix(S) command constructs the matrix of the orthogonal linear projection onto the subspace spanned by the vectors in S.  If $B$ is a maximal, linearly independent subset of $S$ and $M$ is the Matrix whose columns are the Vectors in $B$, then

If the conj option is omitted or provided in either of the forms conjugate or conjugate=true, the projection matrix is constructed using Hermitian transpose operations.  If the conj option is given as conjugate=false, the ordinary transpose is used.

Additional arguments are passed as options to the Matrix constructor which builds the result.

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$S≔\left[⟨1\,2\,3\,4⟩\,⟨4\,3\,2\,1⟩\right]$

$S≔\left[\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]\,\left[\begin{array}{c}4\\ 3\\ 2\\ 1\end{array}\right]\right]$

$P≔\mathrm{ProjectionMatrix}\left(S\right)$

$P≔\left[\begin{array}{cccc}\frac{7}{10}& \frac{2}{5}& \frac{1}{10}& -\frac{1}{5}\\ \frac{2}{5}& \frac{3}{10}& \frac{1}{5}& \frac{1}{10}\\ \frac{1}{10}& \frac{1}{5}& \frac{3}{10}& \frac{2}{5}\\ -\frac{1}{5}& \frac{1}{10}& \frac{2}{5}& \frac{7}{10}\end{array}\right]$

$v≔⟨1\,0\,-1\,3⟩$

$v≔\left[\begin{array}{c}1\\ 0\\ \mathrm{-1}\\ 3\end{array}\right]$

$w≔\mathrm{`.`}\left(P\,v\right)$

$w≔\left[\begin{array}{c}0\\ \frac{1}{2}\\ 1\\ \frac{3}{2}\end{array}\right]$

$\mathrm{Basis}\left(\left[\mathrm{op}\left(S\right)\,w\right]\right)\=S$

$\left[\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]\,\left[\begin{array}{c}4\\ 3\\ 2\\ 1\end{array}\right]\right]=\left[\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]\,\left[\begin{array}{c}4\\ 3\\ 2\\ 1\end{array}\right]\right]$

$\mathrm{`.`}\left(w\,v-w\right)$

$0$

$P≔\mathrm{ProjectionMatrix}\left(\left\{⟨1\,2\,3⟩\,⟨4\,5\,6⟩\right\}\,\mathrm{datatype}\={\mathrm{float}}_8\,\mathrm{shape}\=\mathrm{symmetric}\right)$

$P≔\left[\begin{array}{ccc}0.833333333333333& 0.333333333333333& \mathrm{-0.166666666666667}\\ 0.333333333333333& 0.333333333333333& 0.333333333333333\\ \mathrm{-0.166666666666667}& 0.333333333333333& 0.833333333333333\end{array}\right]$

$\mathrm{`.`}\left(P\,⟨1\,0\,0⟩\right)$

$\left[\begin{array}{c}0.833333333333333\\ 0.333333333333333\\ \mathrm{-0.166666666666667}\end{array}\right]$

$\mathrm{ProjectionMatrix}\left(\left\{⟨a\,1⟩\right\}\right)$

$\left[\begin{array}{cc}\frac{a\stackrel{\&conjugate0;}{a}}{\stackrel{\&conjugate0;}{a}a+1}& \frac{a}{\stackrel{\&conjugate0;}{a}a+1}\\ \frac{\stackrel{\&conjugate0;}{a}}{\stackrel{\&conjugate0;}{a}a+1}& \frac{1}{\stackrel{\&conjugate0;}{a}a+1}\end{array}\right]$

$\mathrm{ProjectionMatrix}\left(\left\{⟨a\,1⟩\right\}\,\mathrm{conjugate}\=\mathrm{false}\right)$

$\left[\begin{array}{cc}\frac{a^2}{a^2+1}& \frac{a}{a^2+1}\\ \frac{a}{a^2+1}& \frac{1}{a^2+1}\end{array}\right]$

The LinearAlgebra[ProjectionMatrix] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.