Two real parts are equal,imaginary partReciprocalcomplexMutualConjugate complex（conjugate complex number）。(When the imaginary part is not equal to 0, it is also called conjugateimaginary number）The conjugate complex number of complex number z is recorded as
(A horizontal line is added to z, which can be read as Conjugate z in English,z conjugate or z bar），Sometimes it can also be expressed as
。
By definition, if z=a+ib (a, b ∈ R), then
=A-ib (a, b ∈ R). On the complex plane, the point corresponding to the conjugate complex number is aboutReal axissymmetric.
Conjugate radical
When
Both are rational roots, and
、
When at least one of them is irrational radical, it is called
As knownhyperbolaThe imaginary axis of isReal axisThe hyperbola whose real axis is the imaginary axis is called the original hyperbolaConjugate hyperbola, such as hyperbola H:
And hyperbola H ':
It is called a pair of conjugate hyperbolas (a>0, b>0).
The main properties are: they have the sameAsymptote, their four focal points are in the same circle, and theirEccentricityThe sum of the squares of the reciprocals of is equal to 1.
Conjugate matrix
Set A=(aij）Is a complex matrix
, where
Indicates aijConjugate complex number of, called
Is the conjugate matrix of A.[2]
Conjugate transpose
After transposing the matrix, replace each number with its conjugate complex number.
Conjugate water depth
Hydraulic jumpMiddle, the mutual or common name of pre jump water depth and post jump water depth.
Conjugate shear joint
Conjugate shear joints exist in structural geology[1]。
definition
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A special relationship between two vectors.Let A be n × n symmetric positive definite matrix, vector p, p ∈R。If the condition (p) Ap=0 is satisfied, then p and p are said to be conjugate directions with respect to A, or p and p are said to be conjugate with respect to A.Generally, for nonzero vector groups p, p,..., p ∈R, if the condition is met: (p) Ap=0 (i ≠ j,i，j=1，2，…，n)，The vector group is said to be about A conjugation.
Conjugate direction method
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A class of descent algorithms that use a set of conjugate directions as search directions to solve unconstrained nonlinear programming problems.Is to study and seek n-ary quadratic function with symmetric positive definite matrix Q
f(x)=1/2xQ x+bx+c
A class of gradient algorithms based on the optimal solution, includingconjugate gradient methodAnd variable scale method.According to the nature of the conjugate direction, a one-dimensional search along a set of directions of Q conjugate can ensure that the minimum point of the quadratic function can be obtained in no more than n steps.Conjugate direction methodIt is also quite effective in dealing with the non quadratic objective function, with superlinear convergence speed, which overcomes the sawtooth phenomenon of the steepest descent method to a certain extent, and avoids the calculation and inversion of Hesse matrix involved in Newton's method.For the non quadratic function, the n-step search cannot obtain the minimum point, so 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 to reconstruct the conjugate direction and continue the search.
