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Proving a matrix is orthogonal

  1. Aug 27, 2016 #1

    joshmccraney

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    1. The problem statement, all variables and given/known data
    Show that the matrix ##P = \big{[} p_{ij} \big{]}## is orthogonal.

    2. Relevant equations
    ##P \vec{v} = \vec{v}'## where each vector is in ##\mathbb{R}^3## and ##P## is a ##3 \times 3## matrix. SO I guess ##P## is a transformation matrix taking ##\vec{v}## to ##\vec{v}'##. I also know ##\vec{v} = v_i \hat{e}_i## where ##\hat{e}_i## is the ##i##th unit vector.

    3. The attempt at a solution
    Orthogonal implies ##P P^t = I##. ##P P^t## can be wrote in component form as ##p_{ij} p_{ji}##. I believe I want to show that ##p_{ij} p_{ji} = \delta_{ij}##. After this I'm not really sure how to proceed. Any ideas?
     
  2. jcsd
  3. Aug 27, 2016 #2
    Hi josh:

    The quoted equation is wrong. You want something similar in which you show the index over which you do the sum required in multiplying a row of P by column of Pt.

    From your problem statement, I am guessing that you were not given a particular matrix you had to show is orthogonal, but rather show a method you can use to show that any given orthogonal matrix is in fact orthogonal. If that is the case, I think your attempted solution (with the correction) is all you need.
    Hope this helps,

    Regards,
    Buzz
     
  4. Aug 27, 2016 #3

    joshmccraney

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    Thanks for taking time to reply Buzz! But when you say

    I don't think I wrote what you quoted? I didn't multiply ##\delta_{ij}## by ##p##. Perhaps you quoted me while I was editing? But I do agree what I wrote was wrong.

    Ok, so to demonstrate ##P## is orthogonal would we have to show ##p_{ki}p_{kj} = \delta_{ij}##?

    And yea, come to think of it I do think ##P## is a general matrix.
     
  5. Aug 28, 2016 #4

    Mark44

    Staff: Mentor

    I'm confused as to what is the actual problem. Did you put part of the problem statement in the relevant equations? If not, the problem statement, as written, is false.
    An arbitrary matrix is not orthogonal.

    Also, what does this mean -- ##\vec{v} = v_i \hat{e}_i##? In the context of vectors in ##\mathbb{R}^3##, it would make more sense to write ##\vec{v} = v_1 \hat{e}_1 + v_2 \hat{e}_2 + v_3\hat{e}_3##
     
  6. Aug 28, 2016 #5

    joshmccraney

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    Yes I did, I'm sorry about that!

    I was using Einstein notation, so it means exactly the sum that you wrote in the end.
     
  7. Aug 28, 2016 #6

    Mark44

    Staff: Mentor

    Wouldn't the right side be shown in brackets, like this?
    ##[ v_i \hat{e}_i]##
    This is similar to the shorthand notation ##[p_{ij}]## that you used in the OP to represent all of the entries of matrix P.

    In any case, what is the exact problem statement? From what you've provided so far, I don't see how one can show that an arbitrary matrix is orthogonal.
     
  8. Aug 28, 2016 #7

    joshmccraney

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    This was also given, but I didn't include it because it seemed like it was of no help:

    If a vector ##\vec{v}## has coordinates ##v_i## with respect to a basis ##\vec{e_i}## , the transformation rule will tell us the coordinates of the same vector ##\vec{v}## with respect to a different basis ##\vec{e_i}'## . Let ##v_i'## denote the coordinates of ##\vec{v}## with respect to ##\vec{e_i}'##. Our goal is to find the transformation rule governing ##v_i'## and ##v_i##.

    Since ##\vec{e_i}## is a basis, it is possible to find a unique set of 9 numbers, ##p_{ij}## such that ##\vec{e_i}' = p_{ij}\vec{e_j}##.
     
  9. Aug 28, 2016 #8

    joshmccraney

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    I have posted the notes that correspond to ##P##.
     
  10. Aug 28, 2016 #9

    Mark44

    Staff: Mentor

    From post #1:
    This also implies that ##P^{-1} = P^t##.

    Maybe I'm missing something, but I don't see anything in the problem description that would lead me to believe that the matrix is orthogonal. You have Pv = v', but you don't show anything about v', other than it is a vector in R3.
     
  11. Aug 28, 2016 #10

    joshmccraney

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    I totally agree. To me it looks as though we are given a square ##3 \times 3## matrix and asked to show this property is true. I'll ask the professor about it, I just wanted to see if anyone else picked up on something I did not. Thanks for your help Mark44!
     
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