Results 1 to 6 of 6
Like Tree2Thanks
  • 1 Post By Idea
  • 1 Post By Idea

Thread: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

  1. #1
    Super Member
    Joined
    Jun 2009
    From
    United States
    Posts
    690
    Thanks
    19

    Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

    We define a linear transformation $\displaystyle T:Mat_{m,m}->Mat_{m,m}$ of $\displaystyle m\times m$ matrices such that $\displaystyle T(M)=M^{Tr}$.

    Basically we are looking at the transposition of a matrix, treating it as a linear transformation. I need to find characteristic polynomials, eigenvalues, eigenvectors, etc. I started by looking at 2-by-2 and 3-by-3 to get some insight. I attached the calculations.
    Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors-pain-ass-problem.png
    Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors-pain-ass-problem2.png

    The 2x2 and 3x3 helped a little, but I still cannot figure out how to generalize it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    1,005
    Thanks
    510

    Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

    $\displaystyle T^2=I$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2009
    From
    United States
    Posts
    690
    Thanks
    19

    Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

    Quote Originally Posted by Idea View Post
    $\displaystyle T^2=I$
    Can you explain a bit? I have read that any matrix such that $\displaystyle A^2=I$ is diagonalizable. I'm not clear how to apply that here.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    1,005
    Thanks
    510

    Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

    the minimal polynomial is $x^2-1$

    so the eigenvalues are $\displaystyle \{-1,1\}$

    yes it is diagonalizable

    the characteristic polynomial is of the form $\displaystyle (x+1)^a(x-1)^b$
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2009
    From
    United States
    Posts
    690
    Thanks
    19

    Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

    Quote Originally Posted by Idea View Post
    the minimal polynomial is $x^2-1$

    so the eigenvalues are $\displaystyle \{-1,1\}$

    yes it is diagonalizable

    the characteristic polynomial is of the form $\displaystyle (x+1)^a(x-1)^b$
    Can you explain you obtained that? I'll try to reproduce your solns but I'm not sure how to start.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    1,005
    Thanks
    510

    Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

    the minimal polynomial of $T$ is the monic polynomial of least degree that $T$ satisfies

    now $T^2(M)=T(M^t)=(M^t)^t=M$ so $T$ satisfies $x^2-1$

    It is clear that $T$ does not satisfy any non zero polynomial of degree less than $2$

    Therefore $x^2-1$ is the minimal polynomial of $T$
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: May 30th 2017, 07:35 AM
  2. how to find a matrix of a linear transformation
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: Jan 28th 2013, 01:00 PM
  3. Find the Matrix of a Linear Transformation
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Dec 9th 2012, 08:13 AM
  4. Linear Transformation of product of two vectors
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Dec 12th 2009, 03:22 AM
  5. Linear transformation of vectors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 26th 2009, 04:08 PM

/mathhelpforum @mathhelpforum