Consequently t22 i is nonsingular so rank a i rank u

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Unformatted text preview: the world did not agree. Hamilton became an unhappy man addicted to alcohol who is reported to have died from a severe attack of gout. Copyright c 2000 SIAM Buy online from SIAM Buy from 510 Chapter 7 Eigenvalues and Eigenvectors 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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