Eigenvalues of a Triangular matrix

**sparky** · February 9th 2011, 01:49 PM

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.

**Chris L T521** · February 9th 2011, 02:05 PM

Originally Posted by sparky

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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