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$A = \left[\begin{array}[c]{rr} a & b\\ c & d\end{array}\right]$

Normally to find the Eigen values of a matrix I would simply find $det(A-λI)$ and equate it to $0$, which in this case gives me $λ^2-(a+d)λ+ad-bc = 0$

With so many variables, how am I supposed to find the Eigen value for this?

mangopancake
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  • Letters from the beginning of the alphabet are often used to denote unspecified constants. So they are not intended as variables, but as numbers to be used in creating a formula. – colormegone Mar 26 '14 at 17:52

1 Answers1

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Use the p-q-formula for quadratic equations :

You get

$$\lambda_1 = \frac{a+d}{2} + \sqrt{\frac{a^2+2ad+d^2}{4}+bc-ad}$$

$$\lambda_2 = \frac{a+d}{2} - \sqrt{\frac{a^2+2ad+d^2}{4}+bc-ad}$$

Peter
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    I'll mention that this is a commonly-presented result in dynamical systems texts and is often written this way. The "trace" of a 2 x 2 matrix is $ \ \text{Tr} \ A \ = \ a + d \ , $ so we write $$ \lambda_{1,2} \ = \ \frac{ \text{Tr} \ A \ \pm \ \sqrt{(\text{Tr} \ A)^2 \ - \ 4 (\text{det} \ A)}}{2} \ \ . $$ – colormegone Mar 27 '14 at 05:51