Complex eigenvalues Brian Krummel April 17, 2020 1 Motivation So far we have focused on computing real eigenvalues and eigenvectors of square matrices. Consider the following example: Example 1. Let A be the rotation matrix by 90 degrees clockwise: 0 −1 A = : 1 0 Find the eigenvalues and eigenvectors of A. Answer. Finding eigenvalues of A. Solving the characteristic equation −λ −1 2 det(A − λI) = = λ + 1 = 0: 1 −λ p Then λ = ±i, where i = −1. In particular, A has complex eigenvalues. 2 Complex numbers In order to find and understand the eigenvalues and eigenvectors of general square matrices, including the rotation matrix in Example 1, we need to introduce complex numbers and complex vectors. p Definition 1. C is the set of all complex numbers z = a + ib, where a; b 2 R and i = −1. One can add and multiply complex numbers z = a + ib and w = c + id by z + w = (a + ib) + (c + id) = (a + c) + i (b + d); zw = (a + ib)(c + id) = ac + i2bd + i (ad + bc) = (ac − bd) + i (ad + bc); where when multiplying zw we used i2 = −1. Also, the complex conjugate z of z = a + ib is z = a − ib: 1 Example 2. If z = 2 + 3i and w = 4 + 5i then z + w = (2 + 3i) + (4 + 5i) = (2 + 4) + i (3 + 5) = 6 + 8i; zw = (2 + 3i) (4 + 5i) = 2 · 4 − 3 · 5 + i (2 · 5 + 3 · 4) = 12 − 15 + i (10 + 12) = −3 + 22i; z = 2 − 3i: Complex conjugation has the following important properties. For each pair of complex numbers z; w 2 C, z + w = z + w; zw = z w: Given a complex number z = a + ib 2 C, we have that z = a − ib is equal to z if and only if b = 0. That is, z = z if and only if z is a real number. Finally, given a complex number z = a + ib 2 C z z = jzj2 = a2 + b2: p We call jzj = a2 + b2 the absolute value or modulus of z. In particular, z z is a non-negative real number. We can represent a complex number z = a+ib as a point (a; b) in R2. Thus we regard z = a+ib as a point in the complex plane, whose axes are the real and imaginary axes. Alternatively, we can represent z as z = reiθ = r cos(θ) + ir sin(θ) is polar coordinates, where z has length r = jzj ≥ 0 and angle θ 2 R (typically 0 ≤ θ < 2π). Imag z r b θ a Real We can express a and b in terms of r and θ usingImag the formulas a = r cos(θ); b = r sin(θ): z Moreover, we can express r and θ in terms of a and rb using the formulas p θ b r = a2 + b2; tan(θ) = : Real a Notice that tan(θ) is defined only up to adding a multiple of π to θ. Hence we need to look at which quadrant of the complex plane z is in to determine the value of θ. 2 p Example 3. Express the complex number z = −8 + 8 3 i in polar coordinates as z = reiθ. Answer. It is useful to first draw a picture with z as a point in the complex plane: Imag 16 8 3 π 3 −8 Real From the picture and the formulas, we see that Imag p p p p r = a2 + b2 = 82 + 82 ·ω3 =i 8 1 + 3 = 8 4 = 8 · 2 = 16; p = 8 3 p tan(θ) = = − 3 ) θ = 2π=3 or 5π=3: −8 Looking at the picture, we seeω that2 z is in 2nd quadrant and thus θ =Real 2π=3. Therefore, z = 16ei2π=3. = − 1 Imag zw1 Geometrically, we can represent complex number addition and multiplication as follows. We add complex vectors z = a + ib and w = c + id asrs ω3 = − i r z z + w = (a + ib) + (c + idϕ) = (a + c) + i (b + d): θ Real Thus we add complex numbers by adding the respective real and imaginary parts. This is just like how we add vectors in R2 by adding their respective components. In fact, C forms a two- dimensional vector space (over the real numbers) with basis f1; ig. Geometrically, we represent complex number addition via the parallelogram rule: Imag z z+w Real w We multiply complex vectors z = reiθ and w = seiφ as zw = reiθ · seiφ = rsei (θ+φ): Thus we multiply complex numbers by multiplying their lengths rs and adding their angles θ + φ. 3 Geometrically, multiplication zw looks like rotating z by φ radians counter-clockwise and scaling the length of z by s: Imag zw rs z ϕ r Imagθ Real a Let k be an integer (or in fact any real number). If weθ multiply z timesReal itself k times to obtain the Imag− k -b k k-th power z , geometrically this looks like scaling kr times by z to get a length r and rotating k times by z to get an angle kθ. In other words, z k z k ikθ k ikθ z = r e = rz+ew : Imag Real w z2 z θ θ θ Real z3 p Example 4. For the complex number z = −8 + 8 3 i, compute z3. Answer. Recall from Example 3 that z = 16ei2π=3. Hence z3 = 163ei 3·2π=3 = 4096ei 2π: Notice that ei 2π = 1 since ei 2π has length one and angle 2π. Therefore, z3 = 4096. Before we move on to complex vectors, let's discuss the meaning of eiθ, where θ 2 R. One interpretation is that z = reiθ simply is a shorthand for polar coordinates. Another interpretation is that eiθ is the exponential function of the complex number iθ. We can define the exponential function ez for a complex number z 2 C using the Taylor series 1 X zk ez = : k! k=0 This series converges absolutely for all z 2 C and thus defines the exponential function ez. Note that the exponential function has many of the same properties for complex numbers as for real 4 numbers; for instance, ez+w = ez · ew for all z; w 2 C. Substituting z = iθ into the Taylor series, we get 1 X (iθ)k eiθ = : k! k=0 Breaking up the sum into two sums where k = 2j is even and k = 2j + 1 is odd, we get 1 1 X (iθ)2j X (iθ)2j+1 eiθ = + : (2j)! (2j + 1)! j=0 j=0 Since i2 = −1, 1 1 X (−1)jθ2j X (−1)jθ)2j+1 eiθ = + i = cos(θ) + i sin(θ); (2j)! (2j + 1)! j=0 j=0 where in the last step we used the Taylor series for cosine and sine. Therefore, eiθ = cos(θ)+i sin(θ) for all θ 2 R. 3 Complex vectors Definition 2. Cn is the set of all n × 1 column vectors whose entries are complex numbers. For instance, 2 1 + i 3 Z = 4 2 − 3i 5 i is a vector in C3. Each of the entries of Z is a complex number. We can express each complex vector Z 2 Cn as Z = X + iY for some real vectors X; Y 2 Rn. We call X the real part of Z and Y the imaginary part of Z. For instance, 2 1 + i 3 2 1 3 2 1 3 4 2 − 3i 5 = 4 2 5 + i 4 −3 5 i 0 1 with 2 1 3 2 1 3 real part = 4 2 5 imaginary part = 4 −3 5 : 0 1 We add and scale complex vectors entry-by-entry like we normally do, except scalars can be complex numbers: 2 1 + i 3 2 4 + 7i 3 2 (1 + i) + (4 + 7i) 3 2 5 + 8i 3 4 2 − 3i 5 + 4 2i 5 = 4 (2 − 3i) + 2i 5 = 4 2 − i 5 i 9 i + 9 9 + i 2 1 + i 3 2 (4 + 5i)(1 + i) 3 2 −1 + 9i 3 (4 + 5i) 4 2 − 3i 5 = 4 (4 + 5i)(2 − 3i) 5 = 4 23 − 2i 5 i (4 + 5i) i −5 + 4i 5 Given Z = X + iY 2 Cn (where X; Y 2 Rn), the complex conjugate of a Z = X + iY is the vector Z = X − iY in Cn. In particular, the entries of Z are the complex conjugates of the corresponding entries of Z. For instance, 2 1 + i 3 2 1 − i 3 conjugate of 4 2 − 3i 5 is 4 2 + 3i 5 : i −i In this course so far we have worked with vector spaces with real number scalars. We could have also developed the theory of vector spaces with complex number scalars. In fact, we could have worked with any scalars in any field. In mathematics, a field F is a set together with addition and multiplication operations that behave just like the real numbers, including subtraction and division; see Wikipedia for the precise definition. One example of a finite field is the integers modulo two Z2 = f0; 1g for which 0+0=0 0+1=1+0=1 1+1=0; 0 · 0 = 0 0 · 1 = 1 · 0 = 0 1 · 1 = 1: Notice in particular that 1+1 = 0. Also note that we could have developed the theory of matrices and linear systems in the complex numbers, or any other field. Definition 3. Let A be an n × n matrix with real or complex entries. We say that λ 2 C is an eigenvalue of A if there is a non-zero vector X 2 Cn such that AX = λX: We call any such non-zero vector X an eigenvector.