Page 298 - Advanced engineering mathematics

278 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

A square matrix is called a diagonal matrix if all the off-diagonal elements are zero. A
diagonal matrix has the appearance

⎛ ⎞
0 0 ··· 0 0
d 1
0 0 ··· 0 0
⎜ d 2 ⎟
⎜ ⎟
0 0 ··· 0 0
⎜ d 3 ⎟
⎜ . . . . .
D = ⎜ ⎟.
⎝ . . . . . . . . . . . ⎟
.
. ⎠
0 0 0 ··· 0 d n
Diagonal matrices have many pleasant properties. Let A and B be n × n diagonal matrices
with diagonal elements, respectively, a ii and b ii .
1. A + B is diagonal with diagonal elements a ii + b ii .
2. AB is diagonal with diagonal elements a ii b ii .
3.
|A|= a 11 a 22 ···a nn ,
the product of the diagonal elements.
4. From (3), A is nonsingular exactly when each diagonal element is nonzero (so A has
nonzero determinant). In this event, A −1 is the diagonal matrix having diagonal elements
1/a ii .
5. The eigenvalues of A are its diagonal elements.
6.
0
⎛ ⎞
⎜ 0 ⎟
⎜ ⎟
⎜.⎟
.
⎜ . ⎟
⎜ ⎟
1
⎜ ⎟
⎜ ⎟
0
⎜ ⎟
⎜ ⎟
⎜ . ⎟
.
⎝ . ⎠
0
with all zero elements except for 1 in the i,1 place, is an eigenvector corresponding to
the eigenvalue a ii .
Most matrices are not diagonal. However, sometimes it is possible to transform a matrix to
a diagonal one. This will enable us to transform some problems to simpler ones.
−1
An n × n matrix A is diagonalizable if there is an n × n matrix P such that P AP is a
diagonal matrix. In this case we say that P diagonalizes A.

We will see that not every matrix is diagonalizable. The following result not only tells us
−1
exactly when A is diagonalizable, but also how to choose P to diagonalize A, and what P AP
must look like.

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October 14, 2010 14:49 THM/NEIL Page-278 27410_09_ch09_p267-294