Hat Operator

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. For example, in the context of errors and residuals, the "hat" over the letter $\hat$ indicates an observable estimate (the residuals) of an unobservable quantity called $\varepsilon$ (the statistical errors).

Another example of the hat 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 $\hat_i = \hat_0+\hat_1 x_i$ where $\sum_i (y_i-\hat_i)^2$ is commonly minimized via least squares by finding optimal values of $\hat_0$ and $\hat_1$ 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:
:$\hat = H \mathbf.$

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.

:$\mathbf \times \mathbf = \mathbf \mathbf$

For example, in three dimensions,

:$\mathbf \times \mathbf = \begin a_x \\ a_y \\ a_z \end \times \begin b_x \\ b_y \\ b_z \end = \begin 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end \begin b_x \\ b_y \\ b_z \end = \mathbf \mathbf$

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 $\hat$ (pronounced "v-hat"). This is especially common in physics context.

Fourier transform

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

Operator

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

$\hat\psi = E\psi$

See also

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References

Mathematical notation

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