Results 1 to 4 of 4

Math Help - Invertible Upper Triangular Matrix

  1. #1
    Junior Member
    Joined
    May 2006
    Posts
    52

    Invertible Upper Triangular Matrix

    Prove that if U is an invertible upper triangular matrix then U^-1 is also upper triangular. How would you prove this?????? TIA
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by TreeMoney View Post
    Prove that if U is an invertible upper triangular matrix then U^-1 is also upper triangular. How would you prove this?????? TIA
    Consider inverting U using Gaussian elimination. We form the augmented
    matrix

    U|I,

    which is obtained by appending the identity matrix on the right of U.

    Then using row operations we reduce the left half of this augmented
    matrix to the identity matrix, and we are left with the inverse of U in
    the right half of the augmented matrix.

    Now if we start from the bottom working our way up with the elimination
    process the right half of the augmented matrix remains upper triangular, hence
    when we finish with the identity in the left half of the augmented matrix
    we have an upper triangular matrix in the right half of the augmented
    matrix. But this upper triangular is the inverse of U, which proves that
    the inverse is upper triangular.

    RonL
    Last edited by CaptainBlack; October 14th 2006 at 01:00 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2006
    Posts
    52
    Thanks RonL. I had no chance of coming up with something like that. Much Appreciated. John
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by TreeMoney View Post
    Thanks RonL. I had no chance of coming up with something like that. Much Appreciated. John
    I got a bit confused about right and left in the original, and have just
    corrected it, you may not have seen the correction.

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Upper triangular matrix
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 14th 2011, 04:34 PM
  2. Replies: 2
    Last Post: January 18th 2011, 08:24 PM
  3. Upper Triangular Matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 20th 2010, 09:06 AM
  4. Upper-Triangular Matrix - Invertible Operator Theorem
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 11th 2010, 04:22 AM
  5. upper triangular matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 14th 2009, 08:34 PM

Search Tags


/mathhelpforum @mathhelpforum