Page 234 - 35Linear Algebra

234 Eigenvalues and Eigenvectors

Figure 12.2: Don’t forget the characteristic polynomial; you will need it to
compute eigenvalues.

det(λI − M) = 0.

The left hand side of this equation is a polynomial in the variable λ
called the characteristic polynomial P M (λ) of M. For an n × n matrix,
the characteristic polynomial has degree n. Then

n
P M (λ) = λ + c 1 λ n−1 + · · · + c n .
n
Notice that P M (0) = det(−M) = (−1) det M.

Now recall the following.

Theorem 12.2.1. (The Fundamental Theorem of Algebra) Any polynomial
can be factored into a product of first order polynomials over C.

This theorem implies that there exists a collection of n complex num-
bers λ i (possibly with repetition) such that

P M (λ) = (λ − λ 1 )(λ − λ 2 ) · · · (λ − λ n ) =⇒ P M (λ i ) = 0.

The eigenvalues λ i of M are exactly the roots of P M (λ). These eigenvalues
could be real or complex or zero, and they need not all be different. The
number of times that any given root λ i appears in the collection of eigenvalues
is called its multiplicity.

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