Section 5.5 Complex Eigenvalues
Motivation: Describe rotations Among transformations, rotations are very simple to describe geometrically. Where are the eigenvectors? A no nonzero vector x is collinear with Ax The corresponding matrix has no real eigenvalues. 0 A = f ( )= +. 0
Complex Numbers Definition The number i is defined such that i =. Now we have to allow all possible combinations a + b i Definition A complex number is a number of the form a + bi for a, b in R. Thesetofall complex numbers is denoted C. ApictureofC uses a plane representation: real axis imaginary axis i i C expands R: realnumbersarewrittena +0i. Vector representation of a + bi: -coordinates What does it mean that a b. 0 and 0 are linearly independent?
Operations on Complex Numbers () Addition: Same as vector addition (a + bi)+(c + di) =(a + b)+(b + d)i. Usually, vectors cannot be multiplied, butcomplexnumberscan! Multiplication: (a + bi)(c + di) =ac +(bc + ad)i + bdi =(ac bd)+(bc + ad)i. Aplanerepresentationofmultiplication of {,, 3,...} by complex z = +i real axis 3i i i 4( ii ) * @ z ( lti ). ( i + i ) r lti ) 3 When z is a real number, multiplication means stretching. When z has an imaginary part, multiplication also means rotation. imaginary axis
Operations on Complex Numbers () The conjugate For a complex number z = a + bi, thecomplex conjugate of z is z = a bi. The following is a convenient definition because: f z = a + bi then zz = (a + bi)(a bi) =a b i = a + b which is a real number! a Note that the length of the vector is p a b + b, There is no geometric interpretation of complex division, but if z 6= 0then: z w = zw ww = zw w. Example: +i i = ( + i) +( ) = +i + i = i.
Notation and Polar coordinates Real and imaginary part: Re(a + bi) =a Absolute value: a + bi = p a + b. m(a + bi) =b. Some properties z + w = z + w zw = z w z = p zz zw = z w Any complex number z = a + bi has the polar coordinates: angle and length. The length is z = p a + b The angle =arctan(b/a) iscalledthe argument of z, andisdenoted =arg(z). The relation with cartesian coordiantes is: z = z (cos + i sin ). {z } unit vector z a z b
More on multiplication t turns out that multiplication has a precise geometric meaning: Complex multiplication Multiply the absolute values and add the arguments: zw = z w arg(zw) =arg(z)+arg(w). zw z + ' z w z w w ' Note arg(z) = arg(z). Multiplying z by z gives a real number because the angles cancel out.
Towards Matrix transformations The point of using complex numbers is to find all eigenvalues of the characteristic polynomial. Fundamental Theorem of Algebra Every polynomial of degree n has exactly n complex roots, countedwith multiplicity. That is, if f (x) =x n + a n x n + + a x + a 0 then f (x) =(x )(x ) (x n) for (not necessarily distinct) complex numbers,,..., n. Conjugate pairs of roots f f is a polynomial with real coe cients, thenthecomplex roots of real polynomials come in conjugate pairs. (Real roots are conjugate of themselves).
The Fundamental Theorem of Algebra Degree and 3 Degree : The quadratic formula gives the (real or complex) roots: f (x) =x + bx + c =) x = b ± p b 4c. For real polynomials, therootsarecomplex conjugates if b Degree 3: 4c is negative. A real cubic polynomial has either three real roots, or one real root and a conjugate pair of complex roots. The graph looks like: or respectively.
The Fundamental Theorem of Algebra Examples Example Degree : p f f ( )= +then p p ± = = p ± i ( ± i) = p. Example Degree 3: Let f ( )=5 3 8 + 0. Since f () = 0, we can do polynomial long division by : We get f ( )=( ) 5 8 +5. Using the quadratic formula, the second polynomial has a root when Therefore, = 8 ± p 64 00 0 f ( )=5( ) = 4 5 ± p 36 0 4+3i 5 = 4 ± 3i. 5 4 3i. 5
Poll The characteristic polynomial of A = p p p is f ( )= +.Thishastwocomplexroots(± i)/. Poll Let s allow vectors with complex entries. What are the eigenvectors of A? the eigenvalue +i p the eigenvalue p i i has eigenvector. i has eigenvector. Do you notice a pattern?
Conjugate Eigenvectors Allowing complex numbers both eigenvalues and eigenvectors of real square matrices occur in conjugate pairs. Conjugate eigenvectors Let A be a real square matrix. f is an eigenvalue with eigenvector v, then is an eigenvalue with eigenvector v. Conjugate pairs of roots in polynomial: f is a root of f,thensois : 0=f ( ) = n + a n n + + a + a 0 = n n + a n + + a + a 0 = f. Conjugate pairs of eigenvectors: Av = =) Av = Av = v = v.
Classification of Matrices with no Real Eigenvalue Triptych p 3+ Pictures of sequence of vectors v, Av, A v,... M = p 3 (in general, a real matrix with not real eigenvalues, depending on the length of eigenvalues). A = p M A = M A = p M p 3 i = p > p 3 = = i p 3 i = p < A 3 v A v Av v v Av A v A 3 v A 3 v A v Av spirals out rotates around an ellipse v spirals in
Picture with Real Eigenvalues Recall the pictures for a matrix with real eigenvalues. Example: Let A = 5 3. 4 3 5 This has eigenvalues =and =,witheigenvectors v = and v =. So A expands the v -direction by and shrinks the v -direction by. v v A v v scale v by scale v by
Picture with Real Eigenvalues We can also draw the sequence of vectors v, Av, A v,... A = 4 5 3 3 5 v Av v A v v A 3 v = > = < Exercise: Draw analogous pictures when, are any combination of <, =,>.