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In mathematics, the conjugate of an expression of the form + is , provided that does not appear in a and b.One says also that the two expressions are conjugate. In particular, the two solutions of a quadratic equation are conjugate, as per the in the quadratic formula =.
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. [1]
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
Conjugate transpose, the complex conjugate of the transpose of a matrix; Harmonic conjugate in complex analysis; Conjugate (graph theory), an alternative term for a line graph, i.e. a graph representing the edge adjacencies of another graph; In group theory, various notions are called conjugation: Inner automorphism, a type of conjugation ...
In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations A x = b . {\displaystyle Ax=b.\,} Unlike the conjugate gradient method , this algorithm does not require the matrix A {\displaystyle A} to be self-adjoint , but instead one needs to perform ...
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, [1] [2] or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle —between them.
In 1912 Dehn gave an algorithm that solves both the word and conjugacy problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2 (the genus 0 and genus 1 cases being trivial). It is known that the conjugacy problem is undecidable for many classes of groups. Classes of group ...
A linear map : gives rise to a corresponding linear map ¯: ¯ ¯ that has the same action as . Note that ¯ preserves scalar multiplication because ¯ = (¯) = ¯ = ¯ Thus, complex conjugation ¯ and ¯ define a functor from the category of complex vector spaces to itself.
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