Page 302 - Advanced engineering mathematics

282 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

Thus far we have proved that, if A has n linearly independent eigenvectors, then A is diag-
−1
onalizable and P AP is the diagonal matrix having the eigenvalues down the main diagonal, in
the order in which the eigenvectors are seen as columns of P.
To prove the converse, now suppose that A is diagonalizable. We want to show that A has
n linearly independent eigenvectors (regardless of whether the eigenvalues are distinct). Further,
−1
we want to show that, if Q AQ is a diagonal matrix, then the diagonal elements of this matrix
are the eigenvalues of A, and the columns of Q are corresponding eigenvectors. Thus suppose
that
0 0
⎛ ⎞
d 1 ···
0 ··· 0
⎜ d 1 ⎟
−1 ⎜ ⎟
Q AQ = ⎜ . . . . ⎟ = D.
⎝ . . . . . . . . ⎠
0 0 ··· d n
Let V j be column j of Q. These columns are linearly independent because Q is nonsingular. We
will show that d j is an eigenvalue of A with eigenvector V j .
−1
From Q AQ=D,wehave AQ=QD. Compute both sides of this equation separately. First,

since the columns of Q are the V j s, then
⎛ ⎞
0 ··· 0
⎞ d 1
⎛
| | ··· |
⎜ 0 d 1 ··· 0 ⎟
⎟
⎜
QD = V 1 V 2 ··· V n ⎠ ⎜ . . . . ⎟
⎝
⎝ . . .
| | ··· | . . . . . ⎠
0 0 ··· d n
⎛ ⎞
| | ··· |
,
= d 1 V 1 d 2 V 2 ··· d n V n ⎠
⎝
| | ··· |
which is a matrix having d j V j as column j. Now compute
⎛ ⎞ ⎛ ⎞
| | ··· | | | ··· |
,
AQ = A V 1 V 2 ··· V n ⎠ = AV 1 AV 2 ··· AV n ⎠
⎝
⎝
| | ··· | | | ··· |
which is a matrix having AV j as column j. Since these matrices are equal, then
AV j = d j V j
and this makes d j an eigenvalue of A with eigenvector V j .
Not every matrix is diagonalizable. We know from the theorem that a n × n matrix with
fewer than n linearly independent eigenvectors is not diagonalizable.
EXAMPLE 9.11
Let

1 −1
B = .
0 1
B has eigenvalues 1,1, and all eigenvectors are constant multiples of

1
.
0
Therefore B has as eigenvectors only nonzero multiples of one vector, and does not have two
linearly independent eigenvectors. By the theorem, B is not diagonalizable.

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October 14, 2010 14:49 THM/NEIL Page-282 27410_09_ch09_p267-294