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!

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

    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,
  2. jcsd
  3. Aug 19, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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?


    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    User Avatar
    Homework Helper

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted