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Expressing Matrix Power as linear combination

  1. Aug 19, 2015 #1
    1. The problem statement, all variables and given/known data

    Okay I am given a matrix A = [2 1 ; 3 4]

    The first step is to find numbers of a and b such that A2 + aA + bI = [0 0; 0 0]

    I is an identity matrix (2x2).

    Part B - After that is says to use the result of the above to express A5 as a linear combination of A and I

    2. Relevant equations

    Okay I am pretty sure for the first part it is just quite simply squaring A, putting the letter a and b in front of the respective matrix and multiplying.
    Than equalling to 0 you have 2 unknowns and 4 equations solve for A and B.
    As for the second part I am not sure if I should be using the characteristic polynomial or/and eigen values?

    3. The attempt at a solution


    Okay so for the first part i got 4 equations once eventually done the computations as:

    2a + b + 17 = 0
    a + 6 = 0
    3a+18=0
    4a+b+19=0

    Solving I get a = -6 and b = 5

    Now for part B I am really stuck. . If i calculate the eigen values then, they are also the eigen values for A5.. Because I + A + A2.... is an infinite series.. with a common ratio... Really stuck sure there is an easier way to look at it,
     
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  3. Aug 19, 2015 #2

    andrewkirk

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    Write ##A^5## as a product of powers of A, each no greater than 2.
    You have an equation that enables you to write ##A^2## as a linear function of A.
    So by repeated substitution for ##A^2## in that product you should be able to get ##A^5## down to a linear function of A.
     
  4. Aug 20, 2015 #3
    Do you mean A2 * A2 * A?

    Sorry could you explain a little more?
     
  5. Aug 20, 2015 #4

    andrewkirk

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    Yes that's what I mean. Now use the equation you derived in Part A to replace each of those two instances of ##A^2## by a linear combination of A and I. Then expand and collect terms. Can you see a way forward from there?
     
  6. Aug 20, 2015 #5
    Okay so I sort of understand where you are going but let me clarify....

    So we know the value of a and the value of b... We also know that for the Matrix A2+a*A + b*I we get it equal to a 2x2 zero's matrix.
    Since we know the values of a and b we can plug those into the equation that I got which was A2 * -6A + 5I = [0 0 ; 0 0]
    So basically do I need to replace A2 with A*A than use that result to multiply again by A and repeat?


    EDIT:

    Okay maybe now I understood, do you mean multiply A1 by a than A2 by a and so on?
     
    Last edited: Aug 20, 2015
  7. Aug 20, 2015 #6

    andrewkirk

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    No. That matrix on the right-hand side of your equation is just the zero matrix. So your equation is ##A^2-6A+5I=0##. Now make ##A^2## the subject of the equation and you'll have a formula for ##A^2## that is a linear function of ##A## and ##I##.
     
  8. Aug 20, 2015 #7
    Okay fantastic now I get you!!

    So the real equation for A2 = 6A - 5I

    So since we have the equation of A2 we can multiply that by another linerar combination of A2 to obtain A4 than again multiply by A to get A5?
     
    Last edited: Aug 20, 2015
  9. Aug 20, 2015 #8

    RUber

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    That is right, but the linear combination means you can only have constants times A and constants times I.
    Continue substituting until all higher powers of A are gone and you will be left with cA -DI = A^5.
    Remember to check your answer with a computer or calculator since repeated multiplications and additions provide many opportunities for errors.
     
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