Lecture Notes in Differential Equations - Bruce E. Shapiro

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In practical terms, computation of the determinant is computationally inefficient,
and there are faster ways to calculate the inverse, such as via
Gaussian Elimination. In fact, determinants and matrix inverses are very
rarely used computationally because there is almost always a better way to
solve the problem, where by better we mean the total number of computations
as measure by number of required multiplications and additions.
Definition 14.28. Let A be a square matrix. Then the eigenvalues of A
are the numbers λ and eigenvectors v such that
Av = λv (14.37)
Definition 14.29. The characteristic equation of a square matrix of
order n is the n th order (or possibly lower order) polynomial
det(A − λI) = 0 (14.38)
Example 14.5. Let A be the square matrix defined in equation 14.31.
Then its characteristic equation is
0 =
∣
1 − λ 0 3
4 5 − λ 0
0 3 1 − λ
∣
(14.39)
= (1 − λ)(5 − λ)(1 − λ) − 0 + 3(4)(3) (14.40)
= 41 − 11λ + 7λ 2 − λ 3 (14.41)
Theorem 14.30. The eigenvalues of a square matrix A are the roots of
its characteristic polynomial.
Example 14.6. Let A be the square matrix defined in equation 14.31.
Then its eigenvalues are the roots of the cubic equation
41 − 11λ + 7λ 2 − λ 3 = 0 (14.42)
The only real root of this equation is approximately λ ≈ 6.28761. There are
two additional complex roots, λ ≈ 0.356196 − 2.52861i and λ ≈ 0.356196 +
2.52861i.
Example 14.7. Let
⎛
2 −2
⎞
3
A = ⎝1 1 1 ⎠ (14.43)
1 3 −1