Mathematik-Online lexicon: Sum and Product of the Eigenvalues of a 2x2-Matrix

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	Mathematik-Online lexicon:
	Sum and Product of the Eigenvalues of a 2x2-Matrix

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The matrix

$\displaystyle A=\left(\begin{array}{rr}a & b\\ c & d\end{array}\right)$

has the characteristic polynomial

$\displaystyle p_A(\lambda)=\lambda^2-(a+d)\lambda+(ad-bc)$

with zeros

$\displaystyle \lambda_{1,2}=\frac{(a+d)\pm\sqrt{(a+d)^2-4(ad-bc)}}{2}\,.$

Forming the sum of the eigenvalues the root expression vanishes and we obtain

$\displaystyle \lambda_1+\lambda_2=\frac{(a+d)+(a+d)}{2}=a+d=\operatorname{Spur}(A)\,.$

Multiplying the two eigenvalues we get by the third binomial formula:

$\displaystyle \lambda_1 \lambda_2=\frac{(a+d)^2-(a+d)^2+4(ad-bc)}{4}=ad-bc=\operatorname{det}(A)\,.$

automatisch erstellt am 25. 6. 2018