**Unformatted text preview: **ormation P−1 AP = B (recall §4.8), the fundamental problem for linear
operators is strictly a matrix issue—i.e., ﬁnd a nonsingular matrix P such that
P−1 AP is as simple as possible. The concept of similarity was ﬁrst introduced
on p. 255, but in the interest of continuity it is reviewed below.
70 While it is helpful to have covered the topics in §§4.7–4.9, much of the subsequent development
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Chapter 7
Eigenvalues and Eigenvectors
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Similarity
Two n × n matrices A and B are said to be similar whenever
there exists a nonsingular matrix P such that P−1 AP = B. The
product P−1 AP is called a similarity transformation on A. • It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] • A Fundamental Problem. Given a square matrix A, reduce it to
the simplest possible form by means of a similarity transformation. D
E Diagonal matrices have the simplest form, so we ﬁrst ask, “Is every square
matrix similar to a diagonal matrix?” Linear algebra and matrix theory would
be simpler subjects if this were true, but it’s not. For example, consider
A= 0
0 1
0 T
H , (7.2.1) and observe that A2 = 0 ( A is nilpotent). If there exists a nonsingular matrix
P such that P−1 AP = D, where D is diagonal, then
D2 = P−1 APP−1 AP = P−1 A2 P = 0 =⇒ D = 0 =⇒ A = 0, IG
R which is false. Thus A, as well as any other nonzero nilpotent matrix, is not similar to a diagonal matrix. Nonzero nilpotent matrices are not the only ones that
can’t be diagonalized, but, as we will see, nilpotent matrices play a particularly
important role in nondiagonalizability.
So, if not all square matrices can be diagonalized by a similarity transformation, what are the characteristics of those that can? An answer is easily derived
by examining the equation
⎞
⎛
λ1 0 · · · 0
⎜ 0 λ2 · · · 0...

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