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 must entail that [math]\displaystyle{ P^2=P }[/math].
  • It can range from being a Symmetric Projection Matrix to being a Non-Symmetric Projection Matrix.
    • … tobe a symmetric matrix if and only if the vector space projection is orthogonal.
  • It can be a Hermitian Projection Matrix iff 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):
  • a Nonsymmetric Projection Matrix, such as [math]\displaystyle{ \begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix} }[/math], which projects onto the line [math]\displaystyle{ y=x }[/math].
• See: Linear Transformation, Idempotence, Graphical Projection.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Projection_(linear_algebra) Retrieved:2016-1-14.
  • In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that . That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). 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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