Symmetry in mathematics
Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.
Let f(x) be a real-valued function of a real variable, then f is even if the following equation holds for all x and -x in the domain of f: Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.
Again, let f be a real-valued function of a real variable, then f is odd if the following equation holds for all x and -x in the domain of f: That is, Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.
The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose.
Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries.
Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
(n factorial) possible permutations of a set of n symbols, it follows that the order (i.e., the number of elements) of the symmetric group Sn is n!.
Alternatively, an rth order symmetric tensor represented in coordinates as a quantity with r indices satisfies The space of symmetric tensors of rank r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of the antisymmetric tensor or alternating form.
An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers.
Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component: is necessarily antisymmetric.
To prove it, consider the matrix element: This is zero, because the two particles have zero probability to both be in the superposition state
Isometries have been used to unify the working definition of symmetry in geometry and for functions, probability distributions, matrices, strings, graphs, etc.
Knowledge of a Line symmetry can be used to simplify an ordinary differential equation through reduction of order.
Symmetries may be found by solving a related set of ordinary differential equations.
In the case of a finite number of possible outcomes, symmetry with respect to permutations (relabelings) implies a discrete uniform distribution.
In the case of a real interval of possible outcomes, symmetry with respect to interchanging sub-intervals of equal length corresponds to a continuous uniform distribution.
There is one type of isometry in one dimension that may leave the probability distribution unchanged, that is reflection in a point, for example zero.
For a "random point" in a plane or in space, one can choose an origin, and consider a probability distribution with circular or spherical symmetry, respectively.
