# Projection Matrix

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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.

## 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.^{[1]}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.

- In linear algebra and functional analysis, a

- ↑ Meyer, pp 386+387