Results 1 to 3 of 3

Math Help - Show that matrix is diagonal

  1. #1
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1

    Show that matrix is diagonal

    Problem: Show that if a matrix \textbf{A} is both triangular and unitary, then it is diagonal.

    -------------------------------------
    My attempt at a solution:

    Since \textbf{A} is unitary, \textbf{A}^*=\textbf{A}^{-1} so, \textbf{A}\textbf{A}^*=\textbf{I}.

    The product of a lower-triangular matrix and a upper-triangular matrix is not the identity.
    -------------------------------------

    I would appreciate it if someone could point me in the direction of a more elegant way of doing this.


    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    We show by induction that the jth column of A is empty except for a non-zero entry in the jth coordinate. Recall that A is unitary if and only if its columns are an orthonormal basis of \mathbb{C}^n. Denote the columns of A by a_1, ..., a_n. Then clearly a_1 is empty except for its first entry (because A is diagonal). Now suppose this holds up to j-1. Then, since <a_j,a_k>=0=a_{jk}a_{kk} for 1 \leq k < j, and a_{kk} \neq 0, we must have a_{jk}=0, and thus a_j is also as described.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1
    Quote Originally Posted by Bruno J. View Post
    Then, since <a_j,a_k>=0=a_{jk}a_{kk} for 1 \leq k < j, and a_{kk} \neq 0, we must have a_{jk}=0, and thus a_j is also as described.
    So a_{kk} \neq 0 because \textbf{A} is diagonal.
    Is a_{jk}a_{kk}=0 for 1 \leq k < j the same as saying that \textbf{A} is diagonal?

    I'm not sure how to translate  <a_{j},a_{k}> into English.

    Thank you for your time.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show this matrix is similar to a diagonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: August 3rd 2011, 01:58 PM
  2. Replies: 3
    Last Post: November 24th 2010, 05:26 PM
  3. unitary and upper triangular matrix => diagonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 10th 2009, 06:52 PM
  4. Diagonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 4th 2008, 12:19 PM
  5. diagonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 13th 2006, 04:09 AM

Search Tags


/mathhelpforum @mathhelpforum