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Thread: Theoretical (Similar Matrix)

  1. #1
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    Theoretical (Similar Matrix)

    Show that:
    A) A is similar to A.
    B) If B is similar to A, then A is similar to B.
    C) If A is similar to B and B is similar to C, then A is similar to C.

    I dont know how to do the theoretical problems in my book. I know that similar means:

    A matrix B is said to be similar to a matrix A if there is a nonsingular matrix P such that
    B = P^-1 A P
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by PensFan10 View Post
    Show that:
    A) A is similar to A.
    B) If B is similar to A, then A is similar to B.
    C) If A is similar to B and B is similar to C, then A is similar to C.

    I dont know how to do the theoretical problems in my book. I know that similar means:

    A matrix B is said to be similar to a matrix A if there is a nonsingular matrix P such that
    B = P^-1 A P

    A is similar to matrix say B iff there exist an invertible matrix p such that

    $\displaystyle P^{-1} A P = B $

    so since the inverse for the identity is the identity so

    $\displaystyle I^{-1} A I = A $ this is true we find the matrix p

    ok second one
    B is similar to A so there exist p such that

    $\displaystyle P^{-1} B P = A $

    $\displaystyle P\cdot P^{-1} B P = P A $

    $\displaystyle B P\cdot P^{-1} = P A \cdot P^{-1} $

    $\displaystyle B=P A P^{-1}$

    let $\displaystyle P^{-1} = S $

    $\displaystyle B = S^{-1} A S $ we find the matrix so A is similar to B try in the last one
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  3. #3
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    So for the last one:

    B = R^-1 C R A = P^-1 B P

    wth substitution I get
    A = P^-1(R^-1 C R) P

    not sure where to go from here. I dont know how to simplify this to get that A is similar to C
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  4. #4
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by PensFan10 View Post
    So for the last one:

    B = R^-1 C R A = P^-1 B P

    wth substitution I get
    A = P^-1(R^-1 C R) P

    not sure where to go from here. I dont know how to simplify this to get that A is similar to C
    in general
    the inverse for the matrix

    $\displaystyle AB $

    is $\displaystyle B^{-1}A^{-1} $

    since

    $\displaystyle AB \cdot B^{-1}A^{-1} = I $

    so in your question let

    $\displaystyle RP = S $

    $\displaystyle S^{-1}= $
    did you get it
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  5. #5
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    yes. thank you very much. now if i can figure out linear transformatiions i will be golden.
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