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# Characteristic equation

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

$det(A-\Phi I_n) = 0$

where In is the Identity matrix.

For example, given a matrix Pyorick:

$P_{yorick} = \begin{bmatrix} 5 & -2 & 6 & -1 \\ 0 & 3 & -8 & 0 \\ 0 & 0 & 5 & 4 \\ 0 & 0 & 0& 1 \end{bmatrix}$

$det(P_{yorick} - \Phi I_4) = det(\begin{bmatrix} 5-\Phi & -2 & 6 & -1 \\ 0 & 3-\Phi & -8 & 0 \\ 0 & 0 & 5-\Phi & 4 \\ 0 & 0 & 0& 1-\Phi \end{bmatrix}) # (5-\Phi)^2(3-\Phi)(1-\Phi)$ This would be the Characteristic Equation for Pyorick: $(5-\Phi)^2(3-\Phi)(1-\Phi) 0$ The resulting polynomial is the Characteristic polynomial.

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