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Sum and Product of the Eigenvalues of a 2x2-Matrix


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 overview

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)\,.
$

see also:


  automatisch erstellt am 25.  6. 2018