Questions tagged [projection-matrices]

Let $~x ∈ E^n = V ⊕ W~$. Then $~x~$ can be uniquely decomposed into $$~x = x_1 + x_2~ \qquad(\text{where $~x_1 ∈ V~$ and $~x_2 ∈ W~$})~.$$ The transformation that maps $~x~$ into $~x_1~$ is called the projection matrix (or simply projector) onto $~V~$ along $~W~$ and is denoted as $~φ~$. This is a linear transformation; that is, $$φ(a_1~y_1 + a_2~y_2) = a_1~φ(y_1) + a_2~φ(y_2)$$ for any $~y_1,~ y_2 ∈ E^n~$. This implies that it can be represented by a matrix. This matrix is called a projection matrix and is denoted by $~P_{~V \cdot W}~$ .

The vector transformed by $~P_{~V \cdot W}~$ (that is, $~x_1 = ~P_{~V \cdot W}~ x$) is called the projection (or the projection vector) of $~x~$ onto $~V~$ along $~W~$.

${}$

Theorem: The necessary and sufficient condition for a square matrix $~P~$ of order $~n~$ to be the projection matrix onto $~V = \text{Sp}(P )~$ along $~W = \text{Ker}(P )~$ is given by $$P^2 = P$$

• A square matrix $~{\displaystyle P}~$ is called an orthogonal projection matrix if $~{\displaystyle P^{2}=P=P^{\mathrm {T} }}~$ for a real matrix, and respectively $~{\displaystyle P^{2}=P=P^{\mathrm {H} }}~$ for a complex matrix, where $~{\displaystyle P^{\mathrm {T} }}~$ denotes the transpose of $~{\displaystyle P}~$ and $~{\displaystyle P^{\mathrm {H} }}~$ denotes the Hermitian transpose of $~{\displaystyle P}~$.

• A projection matrix that is not an orthogonal projection matrix is called an oblique projection matrix.

• In statistics, the projection matrix $~( {\displaystyle \mathbf {P} })~$, sometimes also called the influence matrix or hat matrix $~( {\displaystyle \mathbf {H} })~$, maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes the influence each response value has on each fitted value. The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation.

References:

https://en.wikipedia.org/wiki/Projection_(linear_algebra)

https://en.wikipedia.org/wiki/Projection_matrix

http://optics.szfki.kfki.hu/~psinko/alj/menu/04/nummod/Projection_Matrices.pdf