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Math Help - Similarity and Digonalization in matrices

  1. #1
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    Similarity and Digonalization in matrices

    a) If A and B are invertible matrices show that AB and BA are similar

    b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)

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  2. #2
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    Quote Originally Posted by somestudent2 View Post
    a) If A and B are invertible matrices show that AB and BA are similar
    BA = A^{-1}(AB)A.

    Quote Originally Posted by somestudent2 View Post
    b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)
    If A is diagonalisable then P^{-1}AP = D. What happens when you take the inverse of both sides of that equation?
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  3. #3
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    Hi

    for the second one by taking inverse on both sides we arive at

    P^-1A^-1P = D^-1 ?

    And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?

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  4. #4
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    Quote Originally Posted by somestudent2 View Post
    for the second one by taking inverse on both sides we arive at

    P^-1A^-1P = D^-1 ?

    And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?
    If a diagonal matrix has an inverse then the inverse is also diagonal. So all you need to do is to explain why D is invertible. (Reason: the diagonal elements of D are the eigenvalues of A and, since A is invertible, none of them can be zero.)
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