Results 1 to 2 of 2

Math Help - Matrix that has square root

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    58

    Matrix that has square root

    A matrix A belonging to M_n is a square root of B belonging to M_n if A^2=B. Show that every diagonalizable matrix in M_n has a square root.

    I know that a diagonal matrix has all zeros outside of the diagonal and the diagonal may or maynot have zeros. I want to say that these matrices must be similar as well.

    I am not sure how to prove this. Any help would be great. Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Matrix that has square root

    Quote Originally Posted by page929 View Post
    A matrix A belonging to M_n is a square root of B belonging to M_n if A^2=B. Show that every diagonalizable matrix in M_n has a square root.

    I know that a diagonal matrix has all zeros outside of the diagonal and the diagonal may or maynot have zeros. I want to say that these matrices must be similar as well.

    I am not sure how to prove this. Any help would be great. Thanks!
    Here is the jist of it. If a matrix is diagonalizable then there exists a matrix P such that

    \mathbf{A}=\mathbf{PDP^{-1}} where

    \mathbf{D}= \begin{bmatrix}\lambda_1 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & \cdots & 0 & 0 \\ \vdots & & & 0& 0 \\ \vdots & & & & 0 \\ 0 & 0 \cdots & 0 & 0 & \lambda_n  \end{bmatrix}

    Now define the matrix B as

    \mathbf{D}_B= \begin{bmatrix}\sqrt{\lambda_1} & 0 & \cdots & 0 & 0 \\ 0 & \sqrt{\lambda_2} & \cdots & 0 & 0 \\ \vdots & & & 0& 0 \\ \vdots & & & & 0 \\ 0 & 0 \cdots & 0 & 0 & \sqrt{\lambda_n}  \end{bmatrix}

    Where \mathbf{B}=\mathbf{PD}_B\mathbf{P^{-1}}

    Now we have that

    \mathbf{B}^2=(\mathbf{PD}_B\mathbf{P^{-1}})(\mathbf{PD}_B\mathbf{P^{-1}})=\mathbf{PD}_B^2\mathbf{P^{-1}}=\mathbf{PD}\mathbf{P^{-1}}=\mathbf{A}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Square root inside square root equation
    Posted in the Algebra Forum
    Replies: 3
    Last Post: October 10th 2011, 05:17 PM
  2. The square root of a 2x2 matrix
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: July 10th 2010, 08:37 AM
  3. Find the square root of a 3x3 matrix
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 6th 2009, 06:30 AM
  4. Quasi-Newton method with square root matrix
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: June 12th 2009, 08:46 PM
  5. Replies: 12
    Last Post: November 22nd 2008, 01:41 PM

Search Tags


/mathhelpforum @mathhelpforum