Page 299 - Advanced engineering mathematics

9.2 Diagonalization 279

THEOREM 9.6 Diagonalization of a Matrix

Let A be n ×n. Then A is diagonalizable if and only if A has n linearly independent eigenvectors.
−1
Furthermore, if P is the n × n matrix having these eigenvectors as columns, then P AP is the
n × n diagonal matrix having the eigenvalues of A down its main diagonal, in the order in which
the eigenvectors were chosen as columns of P.
In addition, if Q is any matrix that diagonalizes A, then necessarily the diagonal matrix
−1
Q AQ has the eigenvalues of A along its main diagonal, and the columns of Q must be
eigenvectors of A, in the order in which the eigenvalues appear on the main diagonal of
Q AQ.
−1
We will prove the theorem after looking at three examples.

EXAMPLE 9.8
Let

−14
A = .
0 3
A has eigenvalues −1,3 and corresponding linearly independent eigenvectors

1 1
and .
0 1
Form

11
P = .
01
Determine

1 −1
−1
P = .
0 1
A simple computation shows that

−10
−1
P AP = ,
0 3
a diagonal matrix with the eigenvalues of A on the main diagonal, in the order in which the
eigenvectors were used to form the columns of
If we reverse the order of these eigenvectors as columns and define

11
Q = ,
10
then

3 0
−1
Q AQ =
0 −1
with the eigenvalues along the main diagonal, but now in the order reflecting the order of the
eigenvectors used in forming the columns of Q.

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October 14, 2010 14:49 THM/NEIL Page-279 27410_09_ch09_p267-294