Page 233 - 35Linear Algebra

12.2 The Eigenvalue–Eigenvector Equation 233

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1. Find the characteristic polynomial of the matrix M for L, given by det(λI−M).
2. Find the roots of the characteristic polynomial; these are the eigenvalues of L.

3. For each eigenvalue λ i , solve the linear system (M − λ i I)v = 0 to obtain an
eigenvector v associated to λ i .

Jordan block example

12.2 The Eigenvalue–Eigenvector Equation

In section 12, we developed the idea of eigenvalues and eigenvectors in the
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case of linear transformations R → R . In this section, we will develop the
idea more generally.

Eigenvalues

Definition If L: V → V is linear and for some scalar λ and v 6= 0 V

Lv = λv.

then λ is an eigenvalue of L with eigenvector v.

This equation says that the direction of v is invariant (unchanged) under L.
Let’s try to understand this equation better in terms of matrices. Let V
be a finite-dimensional vector space and let L: V → V . If we have a basis
for V we can represent L by a square matrix M and find eigenvalues λ and
associated eigenvectors v by solving the homogeneous system

(M − λI)v = 0.

This system has non-zero solutions if and only if the matrix

M − λI

is singular, and so we require that

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To save writing many minus signs compute det(M − λI); which is equivalent if you
only need the roots.

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