en>Mogism: /* History */Cleanup/Typo fixing, typo(s) fixed: 1920’s → 1920s using AWB

2013-11-14T00:28:20Z

History: Cleanup/Typo fixing, typo(s) fixed: 1920’s → 1920s using AWB

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{{Other uses|Hat (disambiguation)}}
{{Unreferenced|date=January 2007}}
The '''hat operator''' is a [[mathematical notation]] with various uses in different branches of science and [[mathematics]].

==Hat matrix==
{{Main|hat matrix}}
In statistics, the hat matrix ''H'' projects the observed values '''y''' of response variable to the predicted values '''ŷ''':
:<math>\hat{\mathbf{y}} = H \mathbf{y}.</math>

==Cross product==
In [[screw theory]], one use of the [[circumflex|hat]] operator is to represent the [[cross product]] operation. Since the cross product is a [[linear transformation]], it can be represented as a [[Matrix (mathematics)|matrix]]. The hat operator takes a [[coordinate vector|vector]] and transforms it into its equivalent matrix.

:<math>\mathbf{a} \times \mathbf{b} = \mathbf{\hat{a}} \mathbf{b} </math>

For example, in three dimensions,

:<math>\mathbf{a} \times \mathbf{b} = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} \times \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix} \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \mathbf{\hat{a}} \mathbf{b} </math>

==Unit Vector==
{{Main|unit vector}}

==Estimated Value==

In [[statistics]], the hat is used to denote an [[estimator]] or an estimated value, as opposed to its theoretical counterpart. For example, in the context of [[Errors and residuals in statistics|errors and residuals]], the "hat" over the letter ε indicates an observable estimate (the residuals) of an unobservable quantity called ε (the statistical errors).

==See also==
* [[Exterior algebra]]
* [[Top-hat filter]]

{{DEFAULTSORT:Hat Operator}}
[[Category:Mathematical notation]]

{{Algebra-stub}}

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en>Mogism: /* History */Cleanup/Typo fixing, typo(s) fixed: 1920’s → 1920s using AWB