Linear Algebra/Characteristic Equation

The matrix definition of an eigen value is very useful since it allows us to find eigen values for a given matrix using the following theorem:


 * $$ \lambda $$ is an eigen value of $$A$$ iff $$det(A - \lambda I_n v) = 0. $$

Proof:


 * If $$ Av = \lambda v $$ then

$$ \Rightarrow Av = \lambda I_n v$$

$$ \Rightarrow Av - \lambda I_n v = 0$$

$$ \Rightarrow (A - \lambda I_n) v = 0$$

but since $$ v$$ is non-zero we know that $$ (A - \lambda I_n) $$ is singular, ie it's determinant is zero so an eigen value of A will satisfy the equation


 * $$det(A - \lambda I_n v) = 0. $$

which is known as the characteristic equation. (haven't proved the converse, but this is not required when calculating eigenvalues).

In the case $$ A $$ is a $$ 2x2 $$ matrix, this equation leads to the characteristic polynomial :


 * $$ det( \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} - \lambda \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} ) = 0 $$


 * $$ \Rightarrow det( \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} - \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} ) = 0 $$


 * $$ \Rightarrow det\begin{bmatrix}a_{11}- \lambda & a_{12} \\ a_{21} & a_{22} - \lambda \end{bmatrix} = 0 $$


 * $$ \Rightarrow (a_{11} - \lambda)(a_{22} - \lambda) - a_{21} a_{12} = 0 $$
 * $$ \Rightarrow \lambda^2 - ( a_{11} + a_{22} ) \lambda + a_{11} a_{22} - a_{12} a_{21} = 0  $$

This is simply a quadratic equation and the roots of this are the eigen values of $$ A $$

In order to find the corresponding eigen vectors, we simply solve the equation $$ Av = \lambda v $$ which will be two simultaneous equations. There will in fact be infinitely many solutions to this equation since any scalar multiple of an eigen vector is also an eigen vector.