COMPLEX EIGENVALUES Math 21b, O. KnillNOTATION. Complex numbers are written as z =x + iy = r exp(iφ) = r cos(φ) + ir sin(φ). The realnumber r = |z| is called the absolute value of z,the value φ is the argument and denoted by arg(z).Complex numbers contain the real numbers z =x+i0 as a subset. One writes Re(z) = x and Im(z) =y if z = x + iy.ARITHMETIC. Complex numbers are added like vectors: x + iy + u + iv = (x + u) + i(y + v) and multiplied asz ∗w = (x + iy)(u + iv) = xu −yv + i(yu −xv). If z 6= 0, one can divide 1/z = 1/(x + iy) = (x −iy)/(x2+ y2).ABSOLUTE VALUE AND ARGUMENT. The absolute value |z| =px2+ y2satisfies |zw| = |z| |w|. Theargument satisfies arg(zw) = arg(z) + arg(w). These are direct consequences of the polar representation z =r exp(iφ), w = s exp(iψ), zw = rs exp(i(φ + ψ)).GEOMETRIC INTERPRETATION. If z = x + iy is written as a vectorxy, then multiplication with another complex number w is a dilation-rotation: a scaling by |w| and a rotation by arg(w).THE DE MOIVRE FORMULA. zn= exp(inφ) = cos(nφ) + i sin(nφ) = (cos(φ) + i sin(φ))nfollows directlyfrom z = exp(iφ) but it is magic: it leads for example to formulas like cos(3φ) = cos(φ)3− 3 cos(φ) sin2(φ)which would be more difficult to come by using geometrical or power series arguments. This formula is usefulfor example in integration problems likeRcos(x)3dx, which can be solved by using the above deMoivre formula.THE UNIT CIRCLE. Complex numbers of length 1 have the form z = exp(iφ)and are located on the unit circle. The characteristic polynomial fA(λ) =λ5− 1 of the matrix0 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 11 0 0 0 0has all roots on the unit circle. Theroots exp(2πki/5), for k = 0, . . . , 4 lye on the unit circle.THE LOGARITHM. log(z) is defined for z 6= 0 as log |z| + iarg(z). For example, log(2i) = log(2) + iπ/2.Riddle: what is ii? (ii= ei log(i)= eiiπ/2= e−π/2). The logarithm is not defined at 0 and the imaginary partis define only up to 2π. For example, both iπ/2 and 5iπ/2 are equal to log(i).HISTORY. The struggle with√−1 is historically quite interesting. Nagging questions appeared for examplewhen trying to find closed solutions for roots of polynomials. Cardano (1501-1576) was one of the mathemati-cians who at least considered complex numbers but called them arithmetic subtleties which were ”as refined asuseless”. With Bombelli (1526-1573), complex numbers found some practical use. Descartes (1596-1650) calledroots of negative numbers ”imaginary”.Although the fundamental theorem of algebra (below) was still not proved in the 18th century, and complexnumbers were not fully understood, the square root of minus one√−1 was used more and more. Euler (1707-1783) made the observation that exp(ix) = cos x + i sin x which has as a special case the magic formulaeiπ+ 1 = 0 which relate the constants 0, 1, π, e in one equation.For decades, many mathematicians still thought complex numbers were a waste of time. Others used complexnumbers extensively in their work. In 1620, Girard suggested that an equation may have as many roots as itsdegree in 1620. Leibniz (1646-1716) spent quite a bit of time trying to apply the laws of algebra to complexnumbers. He and Johann Bernoulli used imaginary numbers as integration aids. Lambert used complex numbersfor map projections, d’Alembert used them in hydrodynamics, while Euler, D’Alembert and Lagrange used themin their incorrect proofs of the fundamental theorem of algebra. Euler write first the symbol i for√−1.Gauss published the first correct proof of the fundamental theorem of algebra in his doctoral thesis, but stillclaimed in 1825 that the true metaphysics of the square root of −1 is elusive as late as 1825. By 1831Gauss overcame his uncertainty about complex numbers and published his work on the geometric representationof complex numbers as points in the plane. In 1797, a Norwegian Caspar Wessel (1745-1818) and in 1806 aSwiss clerk named Jean Robert Argand (1768-1822) (who stated the theorem the first time for polynomials withcomplex coefficients) did similar work. But these efforts went unnoticed. William Rowan Hamilton (1805-1865)(who would also discover the quaternions while walking over a bridge) expressed in 1833 complex numbers asvectors.Complex numbers continued to develop to complex function theory or chaos theory, a branch of dynamicalsystems theory. Complex numbers are helpful in geometry in number theory or in quantum mechanics. Oncebelieved fictitious they are now most ”natural numbers” and the ”natural numbers” themselves are in fact themost ”complex”. A philospher who asks ”does√−1 really exist?” might be shown the representation of x + iyasx −yy x. When adding or multiplying such dilation-rotation matrices, they behave like complex numbers:for example0 −11 0plays the role of i.FUNDAMENTAL THEOREM OF ALGEBRA. (Gauss 1799) A polynomial of degree n has exactly n roots.CONSEQUENCE: A n × n MATRIX HAS n EIGENVALUES. The characteristic polynomial fA(λ) = λn+an−1λn−1+ . . . + a1λ + a0satisfies fA(λ) = (λ − λ1) . . . (λ − λn), where λiare the roots of f.TRACE AND DETERMINANT. Comparing fA(λ) = (λ − λn) . . . (λ −λn) with λn−tr(A) + .. + (−1)ndet(A)givestr(A) = λ1+ ··· + λn, det(A) = λ1···λn.COMPLEX FUNCTIONS. The characteristic polynomialis an example of a function f from C to C. The graph ofthis function would live in lC × lC which corresponds to afour dimensional real space. One can visualize the functionhowever with the real-valued function z 7→ |f(z)|. Thefigure to the left shows the contour lines of such a functionz 7→ |f (z)|, where f is a polynomial.ITERATION OF POLYNOMIALS. A topic whichis off this course (it would be a course by itself)is the iteration of polynomials like fc(z) = z2+ c.The set of parameter values c for which the iteratesfc(0), f2c(0) = fc(fc(0)), . . . , fnc(0) stay bounded is calledthe Mandelbrot set. It is the fractal black region in thepicture to the left. The now already dusty object appearseverywhere, from photoshop plugins to decorations. InMathematica, you can compute the set very quickly (seehttp://www.math.harvard.edu/computing/math/mandelbrot.m).COMPLEX NUMBERS IN MATHEMATICA OR MAPLE. In both computer algebra systems, the letter I isused for i =√−1. In Maple, you can ask log(1 + I), in Mathematica, this would be Log[1 + I]. Eigenvalues
View Full Document
Unlocking...