Hat notation

A "hat" (circumflex (ˆ)), placed over a symbol is a mathematical notation with various uses.

Estimated value

In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value.^[1] For example, in the context of errors and residuals, the "hat" over the letter ${\hat {\varepsilon }}$ indicates an observable estimate (the residuals) of an unobservable quantity called $\varepsilon$ (the statistical errors).

Another example of the hat operator denoting an estimator occurs in simple linear regression. Assuming a model of $y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i}$ , with observations of independent variable data $x_{i}$ and dependent variable data $y_{i}$ , the estimated model is of the form ${\hat {y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}x_{i}$ where $\sum _{i}(y_{i}-{\hat {y}}_{i})^{2}$ is commonly minimized via least squares by finding optimal values of ${\hat {\beta }}_{0}$ and ${\hat {\beta }}_{1}$ 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:

{\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

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 ${\hat {\mathbf {v} }}$ (pronounced "v-hat").^[2]^[1] This is especially common in physics context.

Fourier transform

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

Operator

In quantum mechanics, operators are denoted with hat notation. For instance, see the time-independent Schrödinger equation, where the Hamiltonian operator is denoted ${\hat {H}}$ .

${\hat {H}}\psi =E\psi$

References

^ ^a ^b Weisstein, Eric W. "Hat". mathworld.wolfram.com. Retrieved 2024-08-29.
^ Barrante, James R. (2016-02-10). Applied Mathematics for Physical Chemistry: Third Edition. Waveland Press. Page 124, Footnote 1. ISBN 978-1-4786-3300-6.

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[:0-1] Weisstein, Eric W. "Hat". mathworld.wolfram.com. Retrieved 2024-08-29.

[2] Barrante, James R. (2016-02-10). Applied Mathematics for Physical Chemistry: Third Edition. Waveland Press. Page 124, Footnote 1. ISBN 978-1-4786-3300-6.

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