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Matrix norm. In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication.
A special case of the theorem was first proved by Hamilton in 1853 [6] in terms of inverses of linear functions of quaternions. [2] [3] [4] This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. Cayley in 1858 stated the result for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case.
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a ...
Calculus, mathematical analysis, statistics, physics. In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers.
Ceiling function. In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor (x). Similarly, the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈x⌉ or ceil (x). [ 1]
Digamma function. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: [ 1][ 2][ 3] It is the first of the polygamma functions. This function is strictly increasing and strictly concave on , [ 4] and it asymptotically behaves as [ 5] for complex numbers with large modulus ( ) in the sector with ...
The Mittag-Leffler function can be used to interpolate continuously between a Gaussian and a Lorentzian function. In mathematics, the Mittag-Leffler function is a special function, a complex function which depends on two complex parameters and . It may be defined by the following series when the real part of is strictly positive: [ 1][ 2] where ...
Cauchy principal value. In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain.