Hat notation

{{Short description|Mathematical notation}} {{For|Wikipedia math syntax|Help:Displaying a formula#Special characters|selfref=y}}  {{one source |date=May 2024}} 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.<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Hat |url=https://mathworld.wolfram.com/Hat.html |access-date=2024-08-29 |website=mathworld.wolfram.com |language=en}}</ref> For example, in the context of errors and residuals, the "hat" over the letter <math>\hat{\varepsilon}</math> indicates an observable estimate (the residuals) of an unobservable quantity called <math>\varepsilon</math> (the statistical errors).

Another example of the hat denoting an estimator occurs in simple linear regression. Assuming a model of <math>y_i = \beta_0+\beta_1 x_i+\varepsilon_i</math>, with observations of independent variable data <math>x_i</math> and dependent variable data <math>y_i</math>, the estimated model is of the form <math>\hat{y}_i = \hat{\beta}_0+\hat{\beta}_1 x_i</math> where <math>\sum_i (y_i-\hat{y}_i)^2</math> is commonly minimized via least squares by finding optimal values of <math>\hat{\beta}_0</math> and <math>\hat{\beta}_1</math> for the observed data.

==Hat matrix== {{Main|hat matrix}} In statistics, the hat matrix ''H'' projects the observed values '''y''' of response variable to the predicted values '''ŷ''': :<math>\hat{\mathbf{y}} = H \mathbf{y}.</math>

==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.

:<math>\mathbf{a} \times \mathbf{b} = \mathbf{\hat{a}} \mathbf{b} </math>

For example, in three dimensions,

:<math>\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}. </math>

==Unit vector== {{Main|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 <math>\hat {\mathbf {v} }</math> (pronounced "v-hat").<ref>{{Cite book |last=Barrante |first=James R. |url=https://books.google.com/books?id=_IHlCwAAQBAJ&dq=%22Hat%22+math+vectors+-wikipedia&pg=PA124 |title=Applied Mathematics for Physical Chemistry: Third Edition |date=2016-02-10 |publisher=Waveland Press |isbn=978-1-4786-3300-6 |at=Page 124, Footnote 1 |language=en}}</ref><ref name=":0" /> This is especially common in physics context.

== Fourier transform == The Fourier transform of a function <math>f</math> is traditionally denoted by <math>\hat{f}</math>.

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

==See also== * {{Annotated link |Exterior algebra}} * {{Annotated link |Glossary of mathematical symbols}} * {{Annotated link |Top-hat filter}} * {{Annotated link |Circumflex}}

==References== {{reflist}}

Category:Mathematical notation