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The hat operator is a mathematical notation with various uses in different branches of science and mathematics.

Hat matrix
Main article: hat matrix

In statistics, the hat matrix H projects the observed values y of response variable to the predicted values ŷ:

\( \hat{\mathbf{y}} = H \mathbf{y}. \)

Cross product

In screw theory, one use of the hat operator is to represent the cross product operation. Since the cross product is a linear transformation, it can be represented as a matrix. The hat operator takes a vector and transforms it into its equivalent matrix.

\( \mathbf{a} \times \mathbf{b} = \mathbf{\hat{a}} \mathbf{b} \)

For example, in three dimensions,

\( \mathbf{a} \times \mathbf{b} = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} \times \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix} \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \mathbf{\hat{a}} \mathbf{b} \)

Unit Vector
Main article: 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, the "hat" over the letter ε indicates an observable estimate (the residuals) of an unobservable quantity called ε (the statistical errors).

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Hat Operator