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Math Help - Similar Matrices

  1. #1
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    Similar Matrices

    Suppose that A is similar to B. More specifically, suppose that B = P[inverse]AP for some invertible matrix P.

    a. Show that det(A) = det(B)

    b. If v is an eigenvector of A, show that P[inverse] is an eigenvector of B

    c. If A is a diagonalizable matrix, show that B is diagonalizable


    Thanks for all the help!
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  2. #2
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    Quote Originally Posted by Laydieofsorrows View Post
    Suppose that A is similar to B. More specifically, suppose that B = P[inverse]AP for some invertible matrix P.

    a. Show that det(A) = det(B)

    b. If v is an eigenvector of A, show that P[inverse] is an eigenvector of B

    c. If A is a diagonalizable matrix, show that B is diagonalizable


    Thanks for all the help!

    a. You just got to use the properties of determinants:
    B = P^{-1}AP

    det(B) = det(P^{-1}AP)

    det(B) = det(P^{-1})det(A)det(P)

    det(B) = det(A)

    b. Let v be an eigenvector for A and a be the eigenvalue:

    BP^{-1}v=P^{-1}APP^{-1}v=P^{-1}Av=P^{-1}av=aP^{-1}v

    So P^{-1}v is an eigenvector for B, with the same

    eigenvalue a.

    c. Just look at the definition of diagonalizable and remember the product of two invertible matrices is also invertible.
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