Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone Text: Linear Algebra With Applications, Second Edition Otto Bretscher

Monday, Apr 21 Chapter 7.3 Page 324 Problems 16,20.34,36 Main Idea: You want lots of eigen vectors. You might not get all you want. Key Words: Algebraic Multiplicity, Geometric Multiplicity, Eigen Space Goal: Learn to expect additional eigen vectors for multiple roots, but accept graciously the possibility that they do not exist.

Previous Assignment Friday, April 18 Chapter 7.2 Page 310 Problems 6,8,10,20

Page 310 Problem 6 Find all real eigenvalues, with their algebraic multiplicities. A = |1 2 | |3 4 | Det [A-x I ] = Det | 1-x 2 | = (1-x)(4-x) – 6 | 3 4-x | = x 2 -5 x – 2 5 +/- Sqrt [33] x = x = x =

Page 310 Problem 8 Find all real eigenvalues, with their algebraic multiplicities. | | A = | | | |

| -1-x | Det [A-xI] = | x -1 | | x | | -1-x | Det[A-xI] = | x -x 0 | | x | | -1-x | Det [A-xI] = x | | | x |

| -2-x | Det [A-xI] = x | | | x | | -2-x -1 | Det [A-xI] = -x | | | x | Det [A-xI] = -x [ (-2-x)(-1-x) - 2 ] De t[A-xI] = -x [ 2+2x+x+x ] Det [A-xI] = -x [ x 2 + 3x ] Det [A-xI] = -x 2 [ x + 3 ] The eigen values are 0,0,-3.

Page 310 Problem 10 Find all real eigenvalues, with their algebraic multiplicities. | | A = | | | |

| -3-x 0 4 | Det [A - x I ] = | 0 -1-x 0 | | x | | -3-x 0 4 | Det [A - x I ] = (-1-x)| | | x | | -3-x 4 | Det [A - x I ] = (-1-x)| | | -2 3-x |

Det [A - x I ] = (-1-x)( x - 3 x + x ) Det [A - x I ] = (-1-x)( x 2 -1) = -(x+1) 2 (x-1) Eigen values are -1,-1,1

Page 310 Problem 20 Consider a 2x2 matrix A with two distinct real eigenvalues c 1 and c 2. Express Det [A] in terms of c 1 and c 2. Do the same for the trace of A. A = | a b | | c d | Det | a-x b | = (a-x)(d-x) - bc | c d-x |

Det[A-xI] = x 2 -(a+d)x + (ad-bc) = x 2 - trace[A] x + Det[A]. = (x-c 1 )(x-c 2 ) = x 2 - (c 1 +c 2 ) x + c 1 c 2 So c 1 +c 2 = trace[A] and c 1 c 2 = Det[A].

New Material: The Characteristic polynomial of a matrix A is Det[A-xI].

Theorem. The Characteristic polynomial is invariant under similarity.

The Characteristic Polynomial of P -1 A P = Det [ P -1 A P - x I ] = Det [ P -1 (A-x I) P ] = Det [ P -1 Det[A-x I] Det[P] = Det [ P -1 P] Det[A-xI] = Det [A-xI] = The Characteristic Polynomial of A.

Corollary: Similar matrices have the same eigen values. If c, V; are an eigen value and eigen vector of A, then c, P -1 V; are an eigen value and vector of P -1 A P.

Proof: P -1 A P P -1 V = P -1 A V = P -1 c V = c P -1 V.

Definition: The Algebraic multiplicity is the number of times c appears as a root of the characteristic polynomial. The Geometric multiplicity of the eigen value c is the number of linearly independent eigen vectors with eigen value c.

Find the eigen values and eigen vectors of | | A = | | | |

| 0-x 1 0 | Det [A - x I ] = | 0 0-x 1 | = -x 3 | x | The eigen values are 0,0,0.

To find the corresponding eigen vectors. | | | 1 | | | has null space | 0 | | | | 0 | So 0 has algebraic multiplicity 3 and geometric multiplicity 1 for A.

Page 321 Example 7. Solve the recursive dependence relation X(t+1) = A x(t) | 750 | | | Where X(0) = | 200 | and A =(1/20) | | | 200 | | |

The Eigen Values and Vectors are: | 9 | | 2 | | 5 | -3/5 |-12 | -2/5 | -4 | 1 | 4 | | 10 | | 5 | | 2 | | | P = | |. | |

| -3/5 0 0 | P -1.A.P = | 0 -2/5 0 | | | | 750 | | 50 | Solve P X = | 200 | X = | -100 | | 200 | | 100 |

| 9 | | 2 | | 5 | Xo = 50 |-12 | |-4 | | 4 | | 10 | | 5 | | 2 |

| 9 | | 2 | | 5 | X n = 50 (-3/5) n |-12 | (-2/5) n |-4 | +100| 4 | | 10 | | 5 | | 2 | | 500 | The limiting situation is | 400 | | 200 |

Theorem: Eigen Vectors for distinct Eigen Values are linearly independent.

Proof: Suppose that c i, V i are eigen values and eigen vectors for distinct eigen values. Suppose that a 1 V 1 + a 2 V 2 + … + a n V n = 0 then a 1 c 1 n V 1 + a 2 c 2 n V 2 + … + a n c n n V n = 0 for all n. This can only happen when all a i are zero.