Projection Matrix

A projection matrix [math]\displaystyle{ P }[/math] is a [math]\displaystyle{ n \times n }[/math] square matrix that gives a vector space projection from the n-dimensional space [math]\displaystyle{ \R^n }[/math] to a subspace [math]\displaystyle{ W }[/math].

• Context:
  • It can also be stated as: A square matrix [math]\displaystyle{ P }[/math] is a projection matrix if and only if [math]\displaystyle{ P^2=P }[/math]
  • It can be a symmetric matrix if and only if the vector space projection is orthogonal.
  • It can be a Hermitian matrix if and only if the vector space projection satisfies [math]\displaystyle{ \lt v,P_w\gt =\lt v_W,P_w\gt =\lt P_v,w\gt }[/math], where the inner product is the Hermitian inner product. Here [math]\displaystyle{ P_w \in W }[/math] and [math]\displaystyle{ v_W \in W }[/math].
• Example(s):
  • An example of a nonsymmetric projection matrix is [math]\displaystyle{ \begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix} }[/math], which projects onto the line [math]\displaystyle{ y=x }[/math].
• See: Linear Transformation.

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References

2009

• http://en.wikipedia.org/wiki/Projection_(linear_algebra)
  • In linear algebra and functional analysis, a projection is a linear transformation [math]\displaystyle{ P }[/math] from a vector space to itself such that P² = P. It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

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