Hat operator

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

Estimated value[]

In statistics, the hat is used to denote an estimator or an estimated value. 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).

In simple linear regression with observations of independent variable data ${\displaystyle x_{i}}$ and dependent variable data ${\displaystyle y_{i}}$, and assuming a model of ${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i}}$, can lead to an estimated model of the form ${\displaystyle {\hat {y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}x_{i}}$ where ${\displaystyle \sum _{i}(y_{i}-{\hat {y}}_{i})^{2}}$ is minimized via least squares by finding optimal values of ${\displaystyle {\hat {\beta }}_{0}}$ and ${\displaystyle {\hat {\beta }}_{1}}$ for the observed data.

Hat matrix[]

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

${\displaystyle {\hat {\mathbf {y} }}=H\mathbf {y} .}$

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.

${\displaystyle \mathbf {a} \times \mathbf {b} =\mathbf {\hat {a}} \mathbf {b} }$

For example, in three dimensions,

${\displaystyle \mathbf {a} \times \mathbf {b} ={\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}}\times {\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}}={\begin{bmatrix}0&-a_{z}&a_{y}\\a_{z}&0&-a_{x}\\-a_{y}&a_{x}&0\end{bmatrix}}{\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}}=\mathbf {\hat {a}} \mathbf {b} }$

Unit vector[]

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in ${\displaystyle {\hat {\mathbf {v} }}}$ (pronounced "v-hat").^[1]

Fourier transform[]

The Fourier transform of a function ${\displaystyle f}$ is traditionally denoted by ${\displaystyle {\hat {f}}}$.

Quantum mechanics[]

In quantum mechanics, the capital H occurs after you partially differentiate Schrödinger's equation.

See also[]

• Exterior algebra
• Top-hat filter
• Circumflex, noting that precomposed glyphs [letter-with-circumflex] do not exist for all letters.

References[]

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