1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear combination of eigenvs

  1. Jun 8, 2008 #1
    1. The problem statement, all variables and given/known data

    1. Initially, we assume the breeder’s plants are all growing in a field where they will be
    cross-polenated randomly, with genes that can come from anywhere (even neighbouring
    fields of flowers). A given plant would thus be crossed randomly, so that its offspring
    would get an R or W gene with equal probability. As a result the offspring of the plants
    will be as follows:

    the offspring of the red plants will be 50% red, 50% pink and 0% white
    the offspring of the pink plants will be 25% red, 50% pink and 25% white
    the offspring of the white plants will be 0% red, 50% pink and 50% white

    Create a transition matrix A so that, given the state vector ~St at time t, the fraction of
    the breeder’s crop that is of each color the next year (time t + 1) can be found as
    St+1 = A*St
    St= [rt+1 pt+1 wt+1]^T
    Use the probabilities above to populate the columns of the matrix.

    2. It turns out that red flowers are the most popular at the florists, so the breeder begins
    with an initial state vector, S0, with r0 = 1/2, p0 = 1/4, and w0 = 1/4. Using your matrixfrom question 1, determine the proportions of each type of flower in years t = 1, and t = 2.

    3. Write the initial state vector 2 as linear combinations of the eigenvectors.

    2. Relevant equations

    3. The attempt at a solution

    1. A=
    [1/2 1/4 0
    1/2 1/2 1/2
    0 1/4 1/2]

    2. S(t+1)=AS(t)

    so to find S(3) I first need my S(2) and S(1)
    which I found:

    t=0, S(1)= A*S(0)

    [1/2 1/4 0 [1/2 [5/16
    1/2 1/2 1/2 X 1/4 = 1/2
    0 1/4 1/2] 1/4] 3/16]
    *these I know are correct because the proportions add up to 1 or 100%

    and t=1 S(2)=AS(1)

    [1/2 1/4 0 [5/16 [9/32
    1/2 1/2 1/2 X 1/2 = 1/2
    0 1/4 1/2] 3/16] 11/32]

    *Could someone please tell me where Im going wrong? My proportions of each colour do not add up to one

    so for my t=2 S(3)=AS(2) Im scared to do because my flowers are disproportional

    3. So here I found my eigenvalues and associated eigenvectors for my matrix A


    Lambda3= 0

    would I be able to write my linear combination of eigen vectors as:
    r(t)= 1/2 + j+k+m
    p(t)= 1/4 +2j -2m
    w(t)= 1/4 +j-k+m

    Help would be really appreciated. I need to be able to find my S(10) from this linear equation and I dont know how to do it.
  2. jcsd
  3. Jun 8, 2008 #2


    User Avatar
    Science Advisor

    (1/4)(1/2)+ (1/2)(3/16)= 1/8+ 3/32= ? (NOT 11/32!)

  4. Jun 8, 2008 #3
    thank you i must have multiplied the other lines wrong much:)
    i figured out number 3 this thread is SOLVED
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook