Hat operator

Hat operator

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

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} imes mathbf{b} = mathbf{hat{a mathbf{b}

For example, in three dimensions,

:mathbf{a} imes mathbf{b} = egin{bmatrix} a_x \ a_y \ a_z end{bmatrix} imes egin{bmatrix} b_x \ b_y \ b_z end{bmatrix} = egin{bmatrix} 0 & -a_z & a_y \ a_z & 0 & -a_x \ -a_y & a_x & 0 end{bmatrix} egin{bmatrix} b_x \ b_y \ b_z end{bmatrix} = mathbf{hat{a mathbf{b}

See also

* Exterior algebra
* Top-hat filter


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