Page 247 - 35Linear Algebra

13.3 Changing to a Basis of Eigenvectors 247

Definition A matrix M is diagonalizable if there exists an invertible ma-
trix P and a diagonal matrix D such that

D = P −1 MP.

We can summarize as follows.
• Change of basis rearranges the components of a vector by the change
of basis matrix P, to give components in the new basis.

• To get the matrix of a linear transformation in the new basis, we con-
jugate the matrix of L by the change of basis matrix: M 7→ P −1 MP.

If for two matrices N and M there exists a matrix P such that M =
P −1 NP, then we say that M and N are similar. Then the above discussion
shows that diagonalizable matrices are similar to diagonal matrices.

Corollary 13.3.1. A square matrix M is diagonalizable if and only if there
exists a basis of eigenvectors for M. Moreover, these eigenvectors are the
columns of a change of basis matrix P which diagonalizes M.

Reading homework: problem 2

Example 132 Let’s try to diagonalize the matrix

 
−14 −28 −44
M =  −7 −14 −23  .
9 18 29

The eigenvalues of M are determined by
2
3
det(M − λI) = −λ + λ + 2λ = 0.
So the eigenvalues of M are −1, 0, and 2, and associated eigenvectors turn out to be

     
−8 −2 −1
v 1 =  −1  , v 2 =  1  , and v 3 =  −1  .
3 0 1

In order for M to be diagonalizable, we need the vectors v 1 , v 2 , v 3 to be linearly
independent. Notice that the matrix

 
−8 −2 −1

P = v 1 v 2 v 3 =  −1 1 −1 
3 0 1

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