Orthogonal Projection
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An Orthogonal Projection is a Linear Projection that ...
- See: Linear Algebra, Functional Analysis, Linear Transformation, Vector Space, Idempotence, Graphical Projection, Geometry, Point (Geometry).
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/projection_(linear_algebra)#Orthogonal_projection Retrieved:2015-2-2.
- For example, the function which maps the point (x, y, z) in three-dimensional space R3 to the point (x, y, 0) is an orthogonal projection onto the x–y plane. This function is represented by the matrix :[math]\displaystyle{ P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}. }[/math]
The action of this matrix on an arbitrary vector is
:[math]\displaystyle{ P \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \lt P\gt x \\ y \\ 0 \end{pmatrix}. }[/math]
To see that P is indeed a projection, i.e., , we compute
:[math]\displaystyle{ P^2 \begin{pmatrix} x \\ y \\ z \end{pmatrix} = P \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} = \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} = P\begin{pmatrix} x \\ y \\ z \end{pmatrix}. }[/math]
- For example, the function which maps the point (x, y, z) in three-dimensional space R3 to the point (x, y, 0) is an orthogonal projection onto the x–y plane. This function is represented by the matrix :[math]\displaystyle{ P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}. }[/math]