Complex conjugate root theorem

Complex conjugate root theorem

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]

It follows from this (and the fundamental theorem of algebra), that if the degree of a real polynomial is odd, it must have at least one real root.[2] That fact can also be proved by using the intermediate value theorem.

Examples and consequences

• The polynomial x2 + 1 = 0 has roots ±i.
• Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue.
• The polynomial
$x^3 - 7x^2 + 41x - 87\,$
has roots
$3,\, 2 + 5i,\, 2 - 5i,$
and thus can be factored as
$(x - 3)(x - 2 - 5i)(x - 2 + 5i).\,$
In computing the product of the last two factors, the imaginary parts cancel, and we get
$(x - 3)(x^2 - 4x + 29).\,$
The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors.

Corollary on odd-degree polynomials

It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root.[2]

This can be proved as follows.

• Since non-real complex roots come in conjugate pairs, there are an even number of them;
• But a polynomial of odd degree has an odd number of roots;
• Therefore some of them must be real.

This requires some care in the presence of multiple roots; but a complex root and its conjugate do have the same multiplicity (and this lemma is not hard to prove). It can also be worked around by considering only irreducible polynomials; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple roots) must have a real root by the reasoning above.

This corollary can also be proved directly by using the intermediate value theorem.

Simple proof

One proof of the theorem is as follows:[2]

Consider the polynomial

$P(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n$

where all ar are real. Suppose some complex number ζ is a root of P, that is P(ζ) = 0. Then

$a_0 + a_1\zeta + a_2\zeta^2 + \cdots + a_n\zeta^n = 0\,$

which can be put as

$\sum a_r\zeta^r = 0.$

Given the properties of complex conjugation,

$\overline {\sum a_r\zeta^r} = \sum \overline{a_r\zeta^r} = \sum a_r \overline{\zeta^r} = \sum a_r\overline{\zeta}^r.$

Thus it follows that

$a_0 + a_1\overline{\zeta} + a_2\overline{\zeta}^2 + \cdots + a_n\overline{\zeta}^n = \overline{0} = 0.$

Notes

1. ^ Anthony G. O'Farell and Gary McGuire (2002). "Complex numbers". Maynooth Mathematical Olympiad Manual. Logic Press. pp. 104. ISBN 0954426908.  Preview available at Google books
2. ^ a b c Alan Jeffrey (2005). "Analytic Functions". Complex Analysis and Applications. CRC Press. pp. 22–23. ISBN 158488553X.

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Complex conjugate — Geometric representation of z and its conjugate in the complex plane In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magni …   Wikipedia

• Complex number — A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the square root of –1. A complex… …   Wikipedia

• Root of unity — The 5th roots of unity in the complex plane In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially …   Wikipedia

• Conjugate element (field theory) — Conjugate elements redirects here. For conjugate group elements, see Conjugacy class. In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the (other) roots of the minimal polynomial pK …   Wikipedia

• Complex plane — Geometric representation of z and its conjugate in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z. In mathematics …   Wikipedia

• Root (mathematics) — This article is about the zeros of a function, which should not be confused with the value at zero. You may also want information on the Nth roots of numbers instead. In mathematics, a root (or a zero) of a complex valued function f is a member x …   Wikipedia

• Fundamental theorem of algebra — In mathematics, the fundamental theorem of algebra states that every non constant single variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed.Sometimes,… …   Wikipedia

• Almost complex manifold — In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex… …   Wikipedia

• List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

• Perron–Frobenius theorem — In linear algebra, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding… …   Wikipedia