Results 1 to 2 of 2

Math Help - Linear maps and diagonal matrices

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    36

    Linear maps and diagonal matrices

    Let T:V->V be a linear map on a finite-dimensional real vector space.
    Suppose A is the matrix representing T with respect to basis e.
    Show that if A is diagonalizable then every matrix representation of T is diagonalizable.
    With this question there is a hint that you may assume that any other matrix representation of T is of the form (X^-1)AX where X is an invertible matrix.

    My thoughts for this question start with that if A is diagonalizable then there exists an invertible matrix X such that (X^-1)AX is diagonal. But I think I need to show that for all X (X^-1)AX is diagonalizable.
    So I think that if (X^-1)AX is a matrix representation of T with respect to another basis then X must consist of the coefficients of the vectors in this basis as linear combinations of the vectors in e.
    I think after this I have to show that (X^-1)AX has n linearly independent eigenvectors as this would imply diagonalizability but I'm not sure how to get there. Any help would be much appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by kevinlightman View Post
    Let T:V->V be a linear map on a finite-dimensional real vector space.
    Suppose A is the matrix representing T with respect to basis e.
    Show that if A is diagonalizable then every matrix representation of T is diagonalizable.
    With this question there is a hint that you may assume that any other matrix representation of T is of the form (X^-1)AX where X is an invertible matrix.

    My thoughts for this question start with that if A is diagonalizable then there exists an invertible matrix X such that (X^-1)AX is diagonal. But I think I need to show that for all X (X^-1)AX is diagonalizable.


    No: since A is diagonalizable there exist an invertible matrix P and a diagonal matrix D s.t. P^{-1}AP=D .

    Now, if B is ANY other matrix representation of the map T then there exists an invertible matrix Q s.t. A=Q^{-1}BQ , but then:

    D=P^{-1}AP=P^{-1}(Q^{-1}BQ)P=(PQ)^{-1}B(PQ) ...and this means not only that B is also diagonalizable but it will have the very diagonal form as A

    Tonio


    So I think that if (X^-1)AX is a matrix representation of T with respect to another basis then X must consist of the coefficients of the vectors in this basis as linear combinations of the vectors in e.
    I think after this I have to show that (X^-1)AX has n linearly independent eigenvectors as this would imply diagonalizability but I'm not sure how to get there. Any help would be much appreciated
    .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Matrices and linear maps.
    Posted in the Advanced Algebra Forum
    Replies: 15
    Last Post: November 24th 2011, 03:13 PM
  2. Matrices as Linear Maps
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 16th 2011, 11:37 PM
  3. General solution for diagonal zero matrices?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 3rd 2010, 04:54 PM
  4. Field of diagonal matrices (or not...)
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: June 30th 2010, 12:48 PM
  5. Diagonal Matrices from Characteristic Poly
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 7th 2008, 12:57 PM

Search Tags


/mathhelpforum @mathhelpforum