Search results
Results from the WOW.Com Content Network
An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
An involution is non-defective, and each eigenvalue equals , so an involution diagonalizes to a signature matrix. A normal involution is Hermitian (complex) or symmetric (real) and also unitary (complex) or orthogonal (real). The determinant of an involutory matrix over any field is ±1. [4]
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
The Fredholm alternative can be applied to solving linear elliptic boundary value problems. The basic result is: if the equation and the appropriate Banach spaces have been set up correctly, then either (1) The homogeneous equation has a nontrivial solution, or (2) The inhomogeneous equation can be solved uniquely for each choice of data.
Another use is to find the minimum norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. The pseudoinverse is defined for all rectangular matrices whose entries are real or complex numbers. Given a rectangular matrix with real or complex entries ...
The inversion taking any point P (other than O) to its image P ' also takes P ' back to P, so the result of applying the same inversion twice is the identity transformation which makes it a self-inversion (i.e. an involution). [2] [3] To make the inversion a total function that is also defined for O, it is necessary to introduce a point at ...
The skew Hermitian elements form a Lie algebra; If 2 is invertible in the *-ring, then the operators 1 / 2 (1 + *) and 1 / 2 (1 − *) are orthogonal idempotents, [2] called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the *-ring is a field) of symmetric and anti ...
In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). [1] In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply ...