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Math Help - Matrices

  1. #1
    mlg
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    Matrices

    I cannot get my head around this question. I would very much appreciate any help please.
    If we
    are given invertible matrices A, B and P so that A = PB, we can say that A is ‘left equivalent to B’. Prove that ‘left equivalence’ is an ‘equivalence relation’.
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    Re: Matrices

    Quote Originally Posted by mlg View Post
    I cannot get my head around this question. I would very much appreciate any help please.
    If we
    are given invertible matrices A, B and P so that A = PB, we can say that A is ‘left equivalent to B’. Prove that ‘left equivalence’ is an ‘equivalence relation’.
    You need to give a complete definition of "left equivalent".
    Is the matrix $P$ fixed? If not how does it function?
    Thanks from topsquark
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    mlg
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    Re: Matrices

    Thanks.
    This type of maths is new to me.
    I need to study it in more detail.
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    Re: Matrices

    Quote Originally Posted by mlg View Post
    Thanks.
    This type of maths is new to me.
    I need to study it in more detail.
    BUT why can you not type in the exact definition from your text/lecture material?
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  5. #5
    mlg
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    Re: Matrices

    Thanks again.
    It mentions 'prove' in the question.
    Would this suggest that a definition is not sufficient?
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  6. #6
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    Re: Matrices

    Quote Originally Posted by mlg View Post
    Thanks again.
    It mentions 'prove' in the question.
    Would this suggest that a definition is not sufficient?
    Plato is asking you to state what your text/course material is giving you for a definition of "equivalence relation."

    -Dan
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  7. #7
    mlg
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    Re: Matrices

    Thanks Dan.
    My text material is giving me the following:
    Matrix similarity is an equivalence relation and to show it we need to check in turn each the following criteria for equivalence: Reflexive, Symmetric and Transitive.
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    Re: Matrices

    Hi,
    I think you are still missing Plato's point. You need the exact definition of "left equivalence". Almost surely, the definition is:
    Given invertible matrices A and B of the same size, A is left equivalent to B if and only if there is an invertible matrix P with A = PB.

    So now you can prove that left equivalence is an equivalence relation on the set of all invertible $n\times n$ matrices for a given constant size $n$.

    Reflexive: Let A be an invertible $n\times n$ matrix. Then A = IA (I is the identity $n\times n$ matrix). So this shows there is an invertible matrix P with A = PA, namely P = I.

    You should now be able to prove the symmetric and transitive properties.

    Edit: I should have added this thought. For a "meaningful" equivalence relation, let m and n be fixed positive integers and $M(n,m)$ the collection of all $n\times m$ matrices with entries from the reals (or any other field). Then for $A,\,B\in M(n,m)$, A is left equivalent to B if and only if there is an invertible $n\times n$ matrix P with A = PB.
    Last edited by johng; May 24th 2014 at 06:37 AM.
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    mlg
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    Re: Matrices

    Thank you johng.
    I will now study this in more detail.
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    mlg
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    Re: Matrices

    Thanks again johng for your time and effort.
    As I said before, this type of maths is new to me. So it will take me a while to study it in more depth.
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    Re: Matrices

    By the way, you say "If we are given invertible matrices A, B and P so that A = PB, we can say that A is ‘left equivalent to B". That doesn't look right to me because there is no "P" in the conclusion and you appear to be saying that A and B being equivalent depends upon some given matrix "P". I suspect what you really mean is "Two matrices, A and B, are left equivalent if and only if there exist an invertible matrix, P, such that A= PB.

    You need to show:
    1) Reflexive. Can you find an invertible matrix, P such that A= PA?

    2) Symmetric. Suppose there exist an invertible matrix, P, such that A= PB. Can you find an invertible matrix, Q, such that B= QA?

    3) Transitive. Suppose there exist an invertible matrix, P, such that A= PB and an invertible matrix Q such that B= QC. Can you find an invertible matrix, R, such that A= RC?
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    mlg
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    Re: Matrices

    Thank you HallsofIvy.
    I'll now work on your suggestions.
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    Re: Matrices

    I'll give you a big hint for proving symmetry of left-equivalent.

    We will assume A is left-equivalent to B, and to show symmetry we need to prove that B is left-equivalent to A.

    Since A is left-equivalent to B, we know there is an invertible matrix P with A = PB.

    Since P is invertible, P-1 exists, so multiply both sides of the equation A = PB by it.
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  14. #14
    mlg
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    Re: Matrices

    Thanks Deveno.
    I'll go ahead and try it.
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