Eigenvalues and eigenvectors

Let $A = (a_{jk})$ be an $n \times n$ matrix and $X$ a column vector. The equation
$AX = \lambda X \cdots (21)$
where $\lambda$ is a number can be written as
$\displaystyle \left( \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\ \cdots \\ x_n \end{array} \right) = \lambda\left( \begin{array}{c} x_1 \\ x_2 \\ \cdots \\ x_n \end{array} \right) \cdots (22)$
or
$\displaystyle \left. \begin{array}{ccc} (a_{11} - \lambda)x_1 + a_{12}x_2 + \cdots + a_{1n}x_n & = & 0 \\ a_{21}x_1 + (a_{22} - \lambda)x_2 + \cdots + a_{2n}x_n & = & 0 \\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots & = & 0 \\ a_{n1}x_1 + a_{n2}x_2 + \cdots + (a_{nn} - \lambda)x_n & = & 0 \end{array}\right\} \cdots(23)$
The equation (23) will have non-trivial solution if and only if
$\displaystyle \left| \begin{array}{cccc} a_{11} - \lambda & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} - \lambda & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} - \lambda \end{array} \right| = 0 \cdots(24)$
which is a polynomial equation of degree n in $\lambda$ . The roots of this equation are called eigenvalues or characteristic values of the matrix $A$ . Corresponding to each eigenvalue there will be a solution $X \ne 0$ , i.e. a non-trivial solution, which is called an eigenvector or characteristic vector belonging to the eigenvalue. The equation (24) can also be written
$\det(A - \lambda I) = 0 \cdots(25)$
and the equation in $\lambda$ is often called the characteristic equation.