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Arnold's cat map is a particularly well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by a square unimodular matrix having no eigenvalues of absolute value 1. [3] The set of the points with a periodic orbit is dense on the torus. Actually a point is periodic if and only if its coordinates are rational.
In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . [citation needed] Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of ...
The Hunt and RLAB color appearance models use the Hunt–Pointer–Estevez transformation matrix (M HPE) for conversion from CIE XYZ to LMS. [4] [5] [6] This is the transformation matrix which was originally used in conjunction with the von Kries transform method, and is therefore also called von Kries transformation matrix (M vonKries).
The DFT is (or can be, through appropriate selection of scaling) a unitary transform, i.e., one that preserves energy. The appropriate choice of scaling to achieve unitarity is , so that the energy in the physical domain will be the same as the energy in the Fourier domain, i.e., to satisfy Parseval's theorem. (Other, non-unitary, scalings, are ...
Rotation matrix. In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix. rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system.
A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated and each have variance 1. [1] The transformation is called "whitening" because it ...
The von Kries coefficient law in color adaptation describes the relationship between the illuminant and the human visual system sensitivity. [1][2][3] The law accounts for the approximate color constancy in the human visual system. [4] It is the oldest and most widely used law to quantify color adaptation, [5] and is used widely in the field of ...
As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators (Nikolski 1988).