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**Unformatted text preview: **P−1 det (B − λI)det (P) = det (B − λI). IG
R In the context of linear operators, this means that the eigenvalues of a matrix
representation of an operator L are invariant under a change of basis. In other
words, the eigenvalues are intrinsic to L in the sense that they are independent
of any coordinate representation.
Now we can establish the fact that every square matrix can be triangularized
by a similarity transformation. In fact, as Issai Schur (p. 123) realized in 1909,
the similarity transformation always can be made to be unitary. Y
P Schur’s Triangularization Theorem O
C Every square matrix is unitarily similar to an upper-triangular matrix.
That is, for each An×n , there exists a unitary matrix U (not unique)
and an upper-triangular matrix T (not unique) such that U∗ AU = T,
and the diagonal entries of T are the eigenvalues of A.
Proof. Use induction on n, the size of the matrix. For n = 1, there is nothing
to prove. For n > 1, assume that all n − 1 × n − 1 matrices are unitarily similar
to an upper-triangular matrix, and consider an n × n matrix A. Suppose that
(λ, x) is an eigenpair for A, and suppose that x has been normalized so that
x 2 = 1. As discussed on p. 325, we can construct an elementary reﬂector
R = R∗ = R−1 with the property that Rx = e1 or, equivalently, x = Re1
(set R = I if x = e1 ). Thus x is the ﬁrst column in R, so R = x | V , and
RAR = RA x | V = R λx | AV = λe1 | RAV = Copyright c 2000 SIAM λ
0 x∗ AV
V∗ AV . Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com
7.2 Diagonalization by Similarity Transformations
http://www.amazon.com/exec/obidos/ASIN/0898714540 509 Since V∗ AV is n − 1 × n − 1, the induction hypothesis insures that there exists
˜
a unitary matrix Q such that Q∗ (V∗ AV)Q = T is upper triangular. If U =
10
∗
R 0 Q , then U is unitary (because U = U−1 ), and It is illegal to print, duplicate, or distribute this material
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