Characteristic equation

In mathematics, in the field of linear algebra, a scalar <math>\Phi</math> is an eigenvalue of an n-by-n matrix if <math>\Phi</math> satisfies the Characteristic Equation:

<math>det(A-\Phi I_n) = 0 </math>

where I_n is the Identity matrix.

For example, given a matrix P_yorick:

<math>P_{yorick} = \begin{bmatrix} 5 & -2 & 6 & -1 \\ 0 & 3 & -8 & 0 \\ 0 & 0 & 5 & 4 \\ 0 & 0 & 0& 1 \end{bmatrix}</math>

<math>det(P_{yorick} - \Phi I_4) = det(\begin{bmatrix} 5-\Phi & -2 & 6 & -1 \\ 0 & 3-\Phi & -8 & 0 \\ 0 & 0 & 5-\Phi & 4 \\ 0 & 0 & 0& 1-\Phi \end{bmatrix})

(5-\Phi)^2(3-\Phi)(1-\Phi)</math> This would be the Characteristic Equation for P_yorick:

<math>(5-\Phi)^2(3-\Phi)(1-\Phi)

0</math>