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>
The resulting polynomial is the
Characteristic polynomial.
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