KNOWLEDGE ADDA: Eigenvalues of 2×2 matrices by Trace and determinant

An analytic solution for the eigenvalues of 2×2 matrices can be obtained directly from the quadratic formula: if

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

The characteristics polynomial is

$\rm det \begin{bmatrix} a-\lambda & b \\ c & d-\lambda \end{bmatrix}=(a-\lambda)(d-\lambda)-bc=\lambda^2-(a+d)\lambda+(ad-bc)$

so the solutions are

$\lambda = \frac{a + d}{2} \pm \sqrt{\frac{(a + d)^2}{4} + bc - ad} = \frac{a + d}{2} \pm \frac{\sqrt{4bc + (a - d)^2 }}{2}$

Notice that the characteristic polynomial of a 2×2 matrix can be written in terms of the TRACE tr(A) = a + d and determinant det(A) = ad − bc as

${\rm det} \begin{bmatrix} a-\lambda & b \\ c & d-\lambda \end{bmatrix} = {\rm det} \left[ A - \lambda I_{2}\right] = \lambda^2- \lambda {\rm tr}(A)+ {\rm det}(A)$

where I₂ is the 2×2 identity matrix The solutions for the eigenvalues of a 2×2 matrix can thus be written as

$\lambda = \frac{1}{2} \left({\rm tr}(A) \pm \sqrt{{\rm tr}^2 (A) - 4 {\rm det}(A)} \right)$

KNOWLEDGE ADDA

Eigenvalues of 2×2 matrices by Trace and determinant

No comments:

Post a Comment