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: Finding a diagonalizable matrix

  1. Oct 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Let M = [{-1, -6}{3, 8}] ( meaning {row1} {row2}

    Find formulas for the entries of M^n, where n is a positive integer.

    3. The attempt at a solution

    So I found the eigenvalues and eigenvectors; e.value1 = 2, e.value2 = 5, e.vector1 = [{-2}{1}], e.vector2 = [{-1}{1}]. I'm 99% sure these are all correct.

    So I proceed to write it as A = PDP^-1, with P as [{-2, -1}{1, 1}], D as [{2, 0}{0, 5}], and P^-1 as 1/10[{8,6}{-3,-1}].

    So next I believe you're supposed to multiply them all together, with the 2 and 5 in D raised to the power of n. I did this and came up with a huge ugly jumble of numbers, which I entered and were incorrect. Am I doing this right? Or did I make a mistake somewhere?

  2. jcsd
  3. Oct 23, 2009 #2


    Staff: Mentor

    You need to incorporate M in your equation, instead of A. So your equation would be
    M = PDP-1
    Mn =(PDP-1)n
    = (PDP-1)(PDP-1)...(PDP-1)
    = (PDP-1PDP-1...PDP-1)
    = PDnP-1

    All the interior PP-1 products simplify to I, and you're left with what's shown above.

    The idea is that, instead of raising M to the power n, you can raise D to the power n (easy to do, since it is a diagonal matrix), and then multiply that on the left by P and on the right by P-1.

    I haven't checked your work, so once you get a result, verify it by comparing Mn for a small value of n (like 2), to see if it agrees with what you have on the right.
  4. Oct 23, 2009 #3
    I did raise D to the power n, and then multiplied on the left by P and on the right by P^-1, and it gave me a big mess
  5. Oct 23, 2009 #4


    Staff: Mentor

    Did you try it to the power 2? Compare M2 and PD2P-1. If they're not equal, here are some things to check.
    1. Your eigenvectors need to be in the matrix P in the same order that the eigenvalues are in the matrix D. IOW, in the same columns.
    2. Are your eigenvalues and eigenvectors correct? It should be that (A - (eigenvalue_1)I)(eigenvector_1) = 0, and the same for the other eigenvalue/eigenvector pair.
    3. You might have a mistake in your calculation for P-1.
  6. Oct 23, 2009 #5


    User Avatar
    Science Advisor

    Your first problem is that 2 and 5 are NOT eigenvalues of that matrix.
    If 2 were an eigenvalue then we must have
    [tex]\begin{bmatrix}-1 & -6 \\ 3 & -8\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=2\begin{bmatrix}x \\ y\end{bmatrix}[/tex]
    Which gives the two equations -x- 6y= 2x and 3x- 8y= 2y. Those are the same as -3x- 6y= 0 and 3x- 10y= 0. Adding the two equations, -16y= 0 which is satisfied only if y= 0 and then we get x= 0. There is no non-trivial vector for which that is true. The same happens if you try 5 rather than 2.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook