math 307 spring, 2003 hentzel time: 1:10-2:00 mwf room: 1324 howe hall
DESCRIPTION
Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: [email protected] http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, - PowerPoint PPT PresentationTRANSCRIPT
Math 307
Spring, 2003Hentzel
Time: 1:10-2:00 MWFRoom: 1324 Howe Hall
Instructor: Irvin Roy HentzelOffice 432 Carver
Phone 515-294-8141E-mail: [email protected]
http://www.math.iastate.edu/hentzel/class.307.ICN
Text: Linear Algebra With Applications, Second Edition Otto Bretscher
• Friday, April 18 Chapter 7.2
• Page 310 Problems 6,8,10,20
• Main Idea: How do you tell what a matrix is going to do?
• Key Words: Eigen Value, Eigen Vector, Characteristic Polynomial
• Goal: Introduction to eigenvalues and eigen vectors.
• Previous Assignment.
• Page 300 Problem 2
• Let A be an invertible nxn matrix
• and V an eigenvector of A with associated eigen value c
• If V is an eigenvector of A^(-1) ? If So, what is its
• eigenvalue.
• If A stretches V by a factor of c, then A^(-1) must
• shrink V by a factor of 1/c.
• Page 300 Problem 4
• Let A be an invertible nxn matrix and V an eigenvector
• of A with associated eigen value c
• Is V an eigen vector of 7A? IF so, what is the eigenvalue?
• If A streches V by a factor of c, then 7 A stretches
• V by a factor of 7 c.
• Page 300 Problem 6
• If a vector V is an eigenvector of both A and B, is
• V necessarily an eigen vector of AB?
• Let A V = a V and B V = b V.
• A B V = A b V = b A V = ba V
• V is an eigen vector of AB and the eigenvalue is ab.
Page 300 Problem 10 Find all 2x2 matrices for which | 1 | | 2 |
is an eigen vector for eigen value 5
• | a b | | 1 | = | 5 |• | c d | | 2 | |10|
• a+2b = 5• c+2d = 10
• a b c d • 1 2 0 0 5• 0 0 1 2 10• • | a | | -2| | 0 | | 5 |• | b | = b | 1| + d | 0 | + | 0 |• | c | | 0| |-2 | | 10 |• | d | | 0| | 1 | | 0 |
• | -2 b + 5 b |• | -2 d +10 d |
• Check.
• | -2 b + 5 b || 1 | | 5|
• | -2 d +10 d || 2 | = |10 |
• Page 300 Problem 40
• Suppose that V is an eigenvector of the nxn
• matrix A, with eigen value 4. Explain why
• V is an eigenvector of A^2 + 2A + 3 In.
• What is its associated eigenvalue.
• (A^2 + 2 A + 3I)V = A(AV) + 2 AV + 3 V • • = (16+8+3)V
• = 27 V.
• Find the Eigen values and vectors of
• | 2 -1 -1 |
• |-1 2 -1 |
• |-1 -1 2 |
• | 2-x -1 -1 |
• Det[A-xI = | -1 2-x -1 |
• | -1 -1 2-x |
• | 2-x -1 -1 |
• Det[A-xI = |-3+x 3-x 0 |
• | -1 -1 2-x |
• | 2-x -1 -1 |
• Det[A-xI = | -1 2-x -1 |
• | -1 -1 2-x |
• | 2-x -1 -1 |
• Det[A-xI = |-3+x 3-x 0 |
• | -1 -1 2-x |
• | 2-x -1 -1 |
• Det[A-xI =(x-3) | 1 -1 0 |
• | -1 -1 2-x |
• | 2-x -1 -1 |
• Det[A-xI =(x-3) |-1+x 0 1 |
• |-3+x 0 3-x |
• Det[A-xI =(x-3) |-1+x 1 |
• |-3+x 3-x |
• Det[A-xI =(x-3)^2 |-1+x 1 |
• | 1 -1 |
• Det[A-xI =(x-3)^2 (-x)
• The eigen values are 3,3,0• • x=3 A-3I = | -1 -1 -1 |• | -1 -1 -1 |• | -1 -1 -1 |
• RCF(A-3I) = | 1 1 1 |• | 0 0 0 |• | 0 0 0 |
• [V1 V2 ] = | -1 -1 |• | 1 0 |• | 0 1 |
• Check:
• | 2 -1 -1 | | -1 -1 | | -3 -3 | | -1 -1 |
• |-1 2 -1 | | 1 0 | = | 3 0 | = 3| 1 0 |
• |-1 -1 2 | | 0 1 | | 0 3 | | 0 1 |
• x = 0
• | 2 -1 -1 | | 1 1 -2 | | 1 0 -1 |
• |-1 2 -1 | ~ | 0 -3 3 | ~ | 0 1 -1 |
• |-1 -1 2 | | 0 3 -3 | | 0 0 0 |
• | 1 |
• V3 = | 1 |
• | 1 |
• Check:
• | 2 -1 -1 | | 1 | | 0 | | 1 |
• |-1 2 -1 | | 1 | = | 0 | = 0 | 1 |
• |-1 -1 2 | | 1 | | 0 | | 1 |
• Diagonalization:
• -1
• | -1 -1 1 | | 2 -1 -1 | | -1 -1 1 |
• | 1 0 1 | | -1 2 -1 | | 1 0 1 |
• | 0 1 1 | | -1 -1 2 | | 0 1 1 |
• | -1 2 -1 | | 2 -1 -1 | | -1 -1 1 |
• 1/3 | -1 -1 2 | | -1 2 -1 | | 1 0 1 |
• | 1 1 1 | | -1 -1 2 | | 0 1 1 |
• | -1 2 -1 | | -1 -1 1 |
• | -1 -1 2 | | 1 0 1 |
• | 0 0 0 | | 0 1 1 |
• | 3 0 0 |
• | 0 3 0 |
• | 0 0 3 |
• Find a formula for the Fibonacci Numbers.
• fo = 1
• f1 = 1
• f2 = 2
• f3 = 3
• fn = fn-1+fn-2.
• | 0 1 | | fn | = | fn+1 | = | fn+1 |
• | 1 1 | | fn+1 | | fn+fn+1| | fn+2 |
•
• n
• F | 1 | = | fn |
• | 1 | | fn+1 |
• Det[ F-xI ] = | -x 1 | = x^2 - x - 1
• | 1 1-x |
• Let the polynomial factor into (x-a)(x-b) where
• 1+Sqrt[5]• a = -----------• 2
• 1-Sqrt[5]• b = ----------• 2
There exist matrices P such that
P^(-1) F P = | a 0 |
| 0 b |
• F = P | a 0 | P^(-1)
• | 0 b |
• F^n = P | a^n 0 | P^(-1)
• | 0 b^n |
• | fn | = P | a^n 0 | P^(-1)
• | fn+1 | | 0 b^n |
• So we have to compute P.
• | -a 1 | ~ | 1 1-a |
• | 1 1-a | | 0 0 |
• | -b 1 | ~ | 1 1-b |
• | 1 1-b | | 0 0 |
• P = [V1 V2] = |-1+a -1+b | = |-b -a |
• | 1 1 | | 1 1 |
• P^(-1) = 1/(a-b) | 1 a |
• | -1 -b |
• F^n = 1/(a-b) | -b -a | | a^n 0 | | 1 a |
• | 1 1 | | 0 b^n | | -1 -b |
• | n n n n |• | -(a b) + a b a b (a - b ) |• | -------------- -(-------------) |• | a - b a - b• |• | n n 1 + n 1 + n |• | a - b a - b |• | ------- --------------- |• | a - b a - b |
• F^n | 1 | = | fn |• | 1 | | fn+1 |
• | n 1 + n n |• | -(a b) - a b + a b (1 + b) |• | --------------------------------- |• | a - b |• | |• | n 1 + n n |• | a + a - b (1 + b) |• | ------------------------ |• | a - b |
•
• So fn = -a^n b - a^(1+n) b + a b^n (1+b)
• -----------------------------------------
• a-b