Results 1 to 2 of 2

Thread: Eigenvalues of a Triangular matrix

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    118

    Eigenvalues of a Triangular matrix

    How do I answer the question below? Do I start by finding the eigenvalues of a triangular matrix first? If so, what do I do with this values? Any assistance will be appreciated

    Question: Show that the eigenvalues of a triangular matrix are the diagonal elements of the matrix.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    5
    Quote Originally Posted by sparky View Post
    How do I answer the question below? Do I start by finding the eigenvalues of a triangular matrix first? If so, what do I do with this values? Any assistance will be appreciated

    Question: Show that the eigenvalues of a triangular matrix are the diagonal elements of the matrix.
    The key here is to note that the determinant of a triangular matrix is the product of the main diagonal elements.

    So if $\displaystyle a_{ij}$ represents the element in the i-th row and j-th column of the $\displaystyle n\times n$ triangular matrix $\displaystyle A$, then

    $\displaystyle \det(A-\lambda I) =\displaystyle\prod\limits_{i=1}^{n}(a_{ii}-\lambda)$.

    So to find the eigenvalues, solve

    $\displaystyle \det(A-\lambda I) = 0\implies\displaystyle\prod\limits_{i=1}^n(a_{ii}-\lambda)=0$.

    Solving for $\displaystyle \lambda$ will get you the diagonal elements of $\displaystyle A$.

    Does this make sense?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: Jan 18th 2011, 08:24 PM
  2. 4 by 4 triangular matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 12th 2010, 05:31 PM
  3. unitary and upper triangular matrix => diagonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Dec 10th 2009, 06:52 PM
  4. Triangular Matrix
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Sep 27th 2009, 06:50 PM
  5. triangular matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 1st 2009, 06:14 PM

Search Tags


/mathhelpforum @mathhelpforum