Hat notation

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

Estimated value

In statistics, a circumflex (ˆ), nicknamed 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 $\hat{ε}$ indicates an observable estimate (the residuals) of an unobservable quantity called $ε$ (the statistical errors).

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

𝐚 \times 𝐛 = \hat{𝐚} 𝐛

For example, in three dimensions,

𝐚 \times 𝐛 = [\begin{matrix} a_{x} \\ a_{y} \\ a_{z} \end{matrix}] \times [\begin{matrix} b_{x} \\ b_{y} \\ b_{z} \end{matrix}] = [\begin{matrix} 0 & - a_{z} & a_{y} \\ a_{z} & 0 & - a_{x} \\ - a_{y} & a_{x} & 0 \end{matrix}] [\begin{matrix} b_{x} \\ b_{y} \\ b_{z} \end{matrix}] = \hat{𝐚} 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 $\hat{𝐯}$ (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 $\hat{H}$ .

$\hat{H} ψ = E ψ$

References

1 2 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] 1 2 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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