Section 5.5. Complex Eigenvalues

Transcription

1 Section 5.5 Complex Eigenvalues

2 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

3 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?

4 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

5 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.

6 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

7 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.

8 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).

9 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.

10 The Fundamental Theorem of Algebra Examples Example Degree : p f f ( )= +then p p ± = = p ± i ( ± i) = p. Example Degree 3: Let f ( )= Since f () = 0, we can do polynomial long division by : We get f ( )=( ) Using the quadratic formula, the second polynomial has a root when Therefore, = 8 ± p f ( )=5( ) = 4 5 ± p i 5 = 4 ± 3i i. 5

11 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?

12 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.

13 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

14 Picture with Real Eigenvalues Recall the pictures for a matrix with real eigenvalues. Example: Let A = 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

15 Picture with Real Eigenvalues We can also draw the sequence of vectors v, Av, A v,... A = v Av v A v v A 3 v = > = < Exercise: Draw analogous pictures when, are any combination of <, =,>.