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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 is accessible without an understanding of this material. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 506 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 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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