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world did not agree. Hamilton became an unhappy man addicted to alcohol who is reported
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510
Chapter 7
Eigenvalues and Eigenvectors
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This form insures that (T − λ1 I)a1 (T − λ2 I)a2 · · · (T − λk I)ak = 0. The characteristic equation for A is p(λ) = (λ − λ1 )a1 (λ − λ2 )a2 · · · (λ − λk )ak = 0, so
U∗ p(A)U = U∗ (A − λ1 I)a1 (A − λ2 I)a2 · · · (A − λk I)ak U
= (T − λ1 I)a1 (T − λ2 I)a2 · · · (T − λk I)ak = 0, It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] and thus p(A) = 0. Note: A completely diﬀerent approach to the Cayley–
Hamilton theorem is discussed on p. 532. Example 7.2.3 Schur’s theorem is not the complete story on triangularizing by similarity.
By allowing nonunitary similarity transformations, the structure of the uppertriangular matrix T can be simpliﬁed to contain zeros everywhere except on
the diagonal and the superdiagonal (the diagonal immediately above the main
diagonal). This is the Jordan form developed on p. 590, but some of the seeds
are sown here. D
E T
H Multiplicities For λ ∈ σ (A) = {λ1 , λ2 , . . . , λs } , we adopt the following deﬁnitions.
• The algebraic multiplicity of λ is the number of times it is repeated as a root of the characteristic polynomial. In other words,
alg multA (λi ) = ai if and only if (x − λ1 )a1 · · · (x − λs )as = 0 is
the characteristic equation for A. • When alg multA (λ) = 1, λ is called a simple eigenvalue. • The geometric multiplicity of λ is dim N (A − λI). In other
words, geo multA (λ) is the maximal number of linearly independent
eigenvectors associated with λ. • Eigenvalues such that alg multA (λ) = geo multA (λ) are called
semisimple eigenvalues of A. It follows from (7.2.2) on p. 511
that...

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