Search results
Results from the WOW.Com Content Network
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.}
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.
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 number of repeats in that list of each element is the separable degree [L:K(α)] sep. A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of its conjugates in the complex numbers have absolute value at most 1, then α is a root of unity. There are quantitative forms of this, stating more precisely ...
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication:
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.
The letter stands for a vector in , is a complex number, and ¯ denotes the complex conjugate of . [1] More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J {\displaystyle J} (different ...
These operators are direct extensions of their complex analogs: if a and b are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers. The product of a nonzero element with its conjugate is a non-negative real number: