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Math Help - Eigenvalues of a Triangular matrix

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    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.
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    Rhymes with Orange Chris L T521's Avatar
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    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 a_{ij} represents the element in the i-th row and j-th column of the n\times n triangular matrix A, then

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

    So to find the eigenvalues, solve

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

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

    Does this make sense?
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