In Mathematics, What Is a Complex Conjugate?

Conjugation has appeared in mathematics, physics, chemistry, geography and other disciplines. Original intention: The rack on the back of two cows is called a yoke, which makes the two cows walk synchronously. A conjugate is a pair that matches according to a certain law. In popular terms, they are twins. In mathematics, there are conjugate complex numbers, conjugate radicals, conjugate hyperbola, and conjugate matrices.

A special relationship between two vectors. Let A be an n × n symmetric positive definite matrix and vector p, p R. If the condition (p) Ap = 0 is satisfied, then p and p are conjugated with respect to A, or p and p are conjugated with A. In general, for a non-zero vector group p, p, ..., p R , if the condition is satisfied: (p) Ap = 0 (i j, i, j = 1, 2, ..., n), the vector is called Group about A conjugation.

A class of descent algorithms that uses a set of conjugate directions as search directions to solve unconstrained nonlinear programming problems. Is in the search for n-ary quadratic functions with a symmetric positive definite matrix Q

f (x) = 1 / 2xQ x + bx + c

A class of gradient algorithms based on the optimal solution, including conjugate gradient method and variable-scale method. According to the nature of the conjugate direction, a one-dimensional search is sequentially performed along a set of directions for the Q conjugate, which can ensure that the minimum point of the quadratic function is obtained in at most n steps. The conjugate direction method is also quite effective when dealing with non-quadratic objective functions. It has a superlinear convergence speed, which overcomes the zigzag phenomenon of the steepest descent method to a certain extent, while avoiding the sea color involved in Newton's method (Hesse ) Calculation and inversion of matrices. For non-quadratic functions, the minimum point cannot be obtained by n-step search. A restart strategy is required, that is, after every n-dimensional search, if the minimum point has not been obtained, the negative gradient direction is used as the initial direction. Construct the conjugate direction and continue searching.