Mathematics-Online lexicon: Annotation toSum and Product of Eigenvalues

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	Mathematik-Online lexicon: Annotation to
	Sum and Product of Eigenvalues

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Let $\lambda_i$ denote the

not necessarily different eigenvalues of $n\times n$ matrix

. Then the follwing holds true:

$\displaystyle \sum_{i=1}^n \lambda_i = \operatorname{Sp}(A),\quad \prod_{i=1}^n \lambda_i = \operatorname{det}(A) \,.$

Since the characteristic polynomial is determinant we find

$\displaystyle p_A(\lambda)=\operatorname{det}(A-\lambda E)=(-\lambda)^n+\operatorname{Sp}(A)(-\lambda)^{n-1}+\cdots+\operatorname{det}(A)\,.$

On the other hand, this polynomial can be represented as

$\displaystyle p_A(\lambda)=\prod_{i=1}^n (\lambda_i-\lambda)\,.$

By comparison of coefficients we obtain the given identities.

(Authors: Burkhardt/Höllig/Hörner )

automatisch erstellt am 8. 7. 2004