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Math Help - Finding commuting matrices

  1. #1
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    Finding commuting matrices

    The exercise says to find all the matrices that commute with
    [ 1 0]
    [-1 0]

    I do
    [ 1 0] _ [a b] ___ [a b] ___ [a b] _ [ 1 0] __ [a-b 0]
    [-1 0] x [c d] = [-a -b] and [c d] x [-1 0] = [c-d 0].
    (I inserted some underscores so the above reads better. The underscores don't denote anything)

    Since the two resulting matrices must be equal I get a=a-b, b=0, -a=c-d and -b=0.

    At this point I get stuck so I look at the solution which says "for equality it is necessary to have a=a-b, b=0, -a=c-d. thus those matrices that commute with the given matrix are all matrices of the form
    [a 0]
    [c c+a] where a and c can take any real values."

    I don't understand this. How do you get from "a=a-b, b=0, -a=c-d" to
    [a 0]
    [c c+a] ?

    Any ideas? The book (Linear Algebra: With Applications - Gareth Williams - Google Books) doesn't show how to find commuting matrices. I couldn't find any such instructions at the web.

    Thanks
    Last edited by jimtehma; March 31st 2013 at 06:09 PM.
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  2. #2
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    Re: Finding commuting matrices

    Quote Originally Posted by jimtehma View Post
    Since the two resulting matrices must be equal I get a=a-b, b=0, -a=c-d and -b=0.

    I don't understand this. How do you get from "a=a-b, b=0, -a=c-d" to

    [a 0]
    [c c+a] ?
    From b=-b=0, we b=0. Substituting this into a=a-b gives a=a, so we can get rid of this condition, as it tells us absolutely nothing. The restriction -a=c-d is the same as saying d=c+a. Since there are no other restrictions on a and c, they can be anything.

    To summarise, a=a,b=0,c=c,d=a+c. Putting this back into your original matrix \begin{bmatrix}a&b\\ c& d\end{bmatrix} gives the desired matrix \begin{bmatrix}a& 0\\ c& c+a\end{bmatrix}
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  3. #3
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    Re: Finding commuting matrices

    Thanks for the quick reply. I understand the steps. I have a followup question though!
    Quote Originally Posted by Gusbob View Post
    The restriction -a=c-d is the same as saying d=c+a. Since there are no other restrictions on a and c, they can be anything.
    Getting d=c+a from -a=c-d is rather trivial, but why don't we do the same for c and a and get c=d-a and a=d-c and use them in the answer matrix too? And thanks for the using the tex shortcuts. I saw them in the reply quote and used them in my post.
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  4. #4
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    Re: Finding commuting matrices

    You could, it would give you the same solution written in a slightly different way. Using Gusbob's method, you get
    \begin{bmatrix}a & 0 \\ c & c+a\end{bmtatrix}
    Doing it your way, you get
    \begin{bmatrix} a & 0 \\ d- a & d \end{bmatrix}
    and
    \begin{bmatrix} d- c & 0 \\ c & d\end{bmatrix}
    But those are not different sets of matrices.
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  5. #5
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    Re: Finding commuting matrices

    Thanks for the quick reply. I see all your "solution" matrices use only one variation of the -a=c-d condition. Is there a rule that dictates this? Can you write the solution as \begin{bmatrix} d-c & 0 \\ d-a & c+a \end{bmatrix} or \begin{bmatrix}a & 0 \\c & d\end{bmatrix} ? This is what confuses me. Getting to the solution is rather trivial, but I'm not sure how should I write the solution as.
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  6. #6
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    Re: Finding commuting matrices

    Quote Originally Posted by jimtehma View Post
    Thanks for the quick reply. I see all your "solution" matrices use only one variation of the -a=c-d condition. Is there a rule that dictates this?
    Heuristically, you have 3 variables and 1 equation. Fixing one of the variables leaves the other two free. It doesn't matter which one you fix since you get the same information out of it anyways. Your first matrix is valid, though it is likely to be confusing to whomever is reading it. Since any of the two variables completely determine the other (solution space is 2 dimensions), your second matrix is not correct (because you have a,c,d free, essentially neglecting the condition -a = c-d).
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  7. #7
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    Re: Finding commuting matrices

    Thanks Gusbob. That clears things up.
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