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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.}
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]
The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (a + ib, a – ib). It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis.
A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division. Argument φ and modulus r locate a point in the complex plane.
The map χ may either be regarded as a conjugate-linear map from V C to itself or as a complex linear isomorphism from V C to its complex conjugate ¯. Conversely, given a complex vector space W with a complex conjugation χ , W is isomorphic as a complex vector space to the complexification V C of the real subspace
If is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π (X) is the conjugate of −π(X*) for all X in g. For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an matrix made up of complex numbers. For an explanation of the notation used here, we begin by representing complex numbers e i θ {\displaystyle e^{i\theta }} as the rotation matrix, that is,
A vinculum can indicate the complex conjugate of a complex number: 2 + 3 i ¯ = 2 − 3 i {\displaystyle {\overline {2+3i}}=2-3i} Logarithm of a number less than 1 can conveniently be represented using vinculum: