Section 3: Complex numbers

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1 Essentially: Section 3: Complex numbers C (set of complex numbers) up to different notation: the same as R 2 (euclidean plane), (i) Write the real 1 instead of the first elementary unit vector e 1 = (1, 0), and (consequently) write the real α instead of αe 1 = (α, 0) ; (ii) write i (the imaginary unit) instead of the second elementary unit vector e 2 = (0, 1), and (consequently) for β R : write βi instead of βe 2 = (0, β) ; (iii) for α, β R : write α + βi instead of αe 1 + βe 2 = (α, β). So: C = { z = α + βi : α, β R } R ; for z = α + βi one calls α the real part of z and β the imaginary part of z, for short: α = Re(z) and β = Im(z). New: A further algebraic structure of C (see next) Section 3: Complex numbers 1

2 The field of complex numbers: algebraic structure Besides the vector space structure, (α 1 + β 1 i) + (α 2 + β 2 i) = (α 1 α 2 ) + (β 2 + β 2 )i (addition) λ (α + βi) = (α + βi) λ = (λα) + (λβ)i (multipl. by a scalar), one defines: Multiplication in C; reciprocal and division (α 1 + β 1 i) (α 2 + β 2 i) = (α 1 α 2 β 1 β 2 ) + (α 1 β 2 + α 2 β 1 )i Comment: On the left hand side we have formally multiplied out and used i 2 = 1 to obtain the right hand side. If z = α + βi 0 (i.e., not α = β = 0), then 1 z = ( ) 1 α 2 +β α βi, i.e., with this 1 2 z one has 1 z z = 1. So the division of any two complex numbers (but, of course, the denominator must be nonzero) is given by: z 1 1 z 2 = z 1 z 2, if z 2 0. Section 3: Complex numbers / 3.1 Algebraic structure 2

3 Multiplication in C extends the well-known multiplication in R preserving its properties (associativity, commutativity, distributivity): (z 1 z 2 )z 3 = z 1 (z 2 z 3 ), z 1 z 2 = z 2 z 1, (z 1 +z 2 )z 3 = z 1 z 3 +z 2 z 3 Recall: Absolute value z = α 2 + β 2 for z = α + βi. Properties: z 1 + z 2 z 1 + z 2, z 1 z 2 = z 1 z 2. Conjugation: A useful automorphism in C For z = α + βi one defines z = α βi, and calls z the conjugated complex of z. The function z z from C to C is an automorphism, i.e.: z 1 + z 2 = z 1 + z 2 and z 1 z 2 = z 1 z 2. Interesting to note: Re(z) = 1 2 (z + z), Im(z) = 1 2i (z z), zz = z 2 and 1 z = 1 z. z 2 Section 3: Complex numbers / 3.1 Algebraic structure 3

4 The complex exponential function Continuation of the real exponential function exp(x) = e x, x R, to a complex-valued function of a complex variable, exp(z) = e z, z C, by: e z = e α ( cos(β) + sin(β)i ), for all z = α + βi. Using well-known properties of the real exponential function and of the sine and cosine functions we see: Basic property of exp is preserved e z 1+z 2 = e z 1 e z 2 for all z 1, z 2 C. The complex exponential function for purely imaginary arguments: e ϕi = cos(ϕ) + sin(ϕ)i, for any ϕ R, and these complex numbers constitute the unit circle in C (as ϕ varies over R or only over an interval of length 2π). Section 3: Complex numbers / 3.2 Complex exponential function 4

5 Euler representation of a complex number z = α + βi : z = r e ϕi, where r = z and ϕ [0, 2π[ ; (corresponds to polar coordinates of points in the euclidean plane). Note: By the Euler representation geometric interpretation of complex multiplication, via z 1 z 2 = r 1 r 2 e (ϕ 1+ϕ 2 )i, when z j = r j e ϕ ji, j = 1, 2. Complex representations of sine and cosine cos ϕ = Re ( e ϕi) = 1 2 ( e ϕi + e ϕi), sin ϕ = Im ( e ϕi) = 1 2i ( e ϕi e ϕi). Note: e ϕi = cos(ϕ) sin(ϕ)i = cos( ϕ) + sin( ϕ)i = e ϕi. Example: cos 2 ϕ = ( 1 2 ( e ϕi +e ϕi)) 2 ( = 1 4 = ( e 2ϕi + e 2ϕi) = 1 2 cos(2ϕ) ) e 2ϕi +e 2ϕi +2 }{{} e 0 = 1 Section 3: Complex numbers / 3.2 Complex exponential function 5

6 Example: One-dim. harmonic oscillation, complex representation s(t) = a cos(ωt + α), t a real variable (time), a > 0 amplitude, ω > 0 ang. frequency, α [0, 2π[ phase shift. Complex representation: S(t) = A e ωti, A = ae αi the complex amplitude. We have: s(t) = Re ( S(t) ), since: a cos(ωt + α) = Re ( ae (ωt+α)i) = Re ( } ae {{ αi } e ωti) = A Section 3: Complex numbers / 3.2 Complex exponential function 6

7 Example: Superposition of two harmonic oscillations with same frequency but different amplitudes and phases: s 1 (t) = a cos(ωt + α), s 2 (t) = b cos(ωt + β). Superposition: s(t) = s 1 (t) + s 2 (t). Use complex representations: S 1 (t) = Ae ωti, A = ae αi ; S 2 (t) = Be ωti, B = be βi. So: s(t) = Re ( S(t) ), where S(t) = S 1 (t) + S 2 (t) = Ae ωti + Be ωti = (A + B)e ωti. Find Euler representation: C = A + B = c e γi. Then: s(t) = Re ( S(t) ) = c cos(ωt + γ). Section 3: Complex numbers / 3.2 Complex exponential function 7

8 Complex polynomials Well-known: A real polynomial of degree n P (x) = c n x n + c n 1 x n c 1 x + c 0, x R, (with real coefficients c 0, c 1,..., c n, where c n 0) may not have any zero, i.e., there may not exist an x 0 R with P (x 0 ) = 0. Example: A second degree polynomial P (x) = x 2 + px + q, where p, q R are such that = ( p 2 2) q < 0, does not have any (real) zeros. However, for a complex polynomial of degree n, P (z) = c n z n + c n 1 z n c 1 z + c 0, z C, (with complex coefficients c 0, c 1,..., c n, where c n 0) the situation is completely different. Section 3: Complex numbers / 3.3 Complex polynomials 8

9 Fundamental theorem on complex polynomials If P (z) is a complex polynomial of degree n 1, then there exists a zero of P (z), i.e., a z 0 C such that P (z 0 ) = 0 ; and, moreover: There exist z 1, z 2,..., z n C such that P (z) factorizes to P (z) = c n (z z 1 ) (z z 2 ) (z z n ), for all z C, (where c n 0 is the highest coefficient of P (z) ). Note: The complex numbers z 1, z 2,..., z n (the zeros of P (z)) need not be all distinct. Example: The above 2nd degree polynomial, but now with a complex variable: P (z) = z 2 + pz + q, z C, where p, q R such that = ( p 2 2) q < 0. Not difficult to check: P (z) = (z z 1 ) (z z 2 ), where z 1,2 = p 2 ± i. Section 3: Complex numbers / 3.3 Complex polynomials 9

10 Complex eigenvalues and eigenvectors For a (real) n n matrix A consider its characteristic polynomial P A (x) and replace the real variable x by a complex variable z. So: Consider the complex characteristic polynomial P A (z). By the fundamental theorem on complex polynomials: P A (z) = ( 1) n (z λ 1 ) (z λ 2 ) (z λ n ), with some complex numbers λ 1, λ 2,..., λ n (not necessarily all distinct). These λ 1, λ 2,..., λ n are called the (complex) eigenvalues of A. Note: Any real eigenvalue of A (as discussed in Section 2) is among the complex eigenvalues λ 1, λ 2,..., λ n. If A is diagonalizable then all the complex eigenvalues λ 1, λ 2,..., λ n are real. In particular, all the eigenvalues λ 1, λ 2,..., λ n of a symmetric matrix A are real. Section 3: Complex numbers / 3.4 Complex eigenvalues and eigenvectors 10

11 Complex eigenvector If λ C is an eigenvalue of the (real) n n matrix A (i.e., if λ is one of the above λ 1, λ 2,..., λ n ), then there exists a complex vector z C n 1, z 0, such that Az = λz. Remark: For a complex column vector z = z 1 z 2. z n C n 1 multiplication with a complex number λ is as usual defined by coordinatewise multiplication; the matrix-vector product Az is formally defined as for real vectors: So Az is the complex n 1 vector with coordinates n a ij z j, (i = 1, 2,..., n). j=1 Section 3: Complex numbers / 3.4 Complex eigenvalues and eigenvectors 11