# Algebraic Multiplicity of Eigenvalues of Linear Operators by Julián López-Gómez

By Julián López-Gómez

This publication brings jointly all on hand effects in regards to the idea of algebraic multiplicities, from the main vintage effects, just like the Jordan Theorem, to the latest advancements, just like the strong point theorem and the development of the multiplicity for non-analytic households. half I (first 3 chapters) is a vintage path on finite-dimensional spectral thought, half II (the subsequent 8 chapters) provides the main common effects on hand concerning the lifestyles and area of expertise of algebraic multiplicities for actual non-analytic operator matrices and households, and half III (last bankruptcy) transfers those effects from linear to nonlinear research. The textual content is as self-contained as attainable and appropriate for college students on the complex undergraduate or starting graduate level.

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22 Chapter 1. The Jordan Theorem and the set Bj := {ej1 , (A − λj I)ej1 , ej2 , (A − λj I)ej2 , . . , ejhν(λ j )−1 , (A − λj I)ejhν(λ j )−1 , ejhν(λ j )−1 j +1 , . . , ehν(λj )−2 } provides us with a basis of N [(A − λj I)2 ]. Moreover, since Aeji = λj eji + (A − λj I)eji , 1 ≤ i ≤ hν(λj )−1 , A(A − λj I)eji = λj (A − λj I)eji , 1 ≤ i ≤ hν(λj )−1 , Aeji = λj eji , hν(λj )−1 + 1 ≤ i ≤ hν(λj )−2 , the matrix of Aj with respect to the basis Bj is the following: Aj,Bj = diag{Jj,2 , Jj,2 , . . , Jj,2 , Jj,1 , Jj,1 , .

The Jordan canonical form 21 Then, hk = dim N [(A − λj I)k+1 ] − dim N [(A − λj I)k ] ≥ 1. j Let {ej1 , . . , ejhν(λ )−1 } be a basis of Cν(λ , and consider the images of the elej )−1 j ments of this basis under the action of the operator A − λj I, {(A − λj I)eji : 1 ≤ i ≤ hν(λj )−1 }. 28) j These images form a linearly independent system of Cν(λ . Indeed, suppose j )−2 there exist ci ∈ C, for 1 ≤ i ≤ hν(λj )−1 , for which hν(λj )−1 ci (A − λj I)eji = 0. i=1 Then, since ν(λj ) − 1 ≥ 1, we have that hν(λj )−1 ci eji ∈ N [A − λj I] ⊂ N [(A − λj I)ν(λj )−1 ] i=1 and, hence, since ej1 , .

Then p (A − λj I)ν(λj ) = 0. j=1 Proof. Let B = {e1 , . . , eN } be a basis of CN . Then, for each 1 ≤ i ≤ N , the N + 1 vectors 0 ≤ j ≤ N, Aj ei , must be linearly dependent, and, hence, there exist (αi0 , . . , αiN ) ∈ CN +1 \ {0} 10 Chapter 1. The Jordan Theorem such that N αij Aj ei = 0. j=0 Now, consider the polynomials Q1 , . . , QN deﬁned by N z ∈ C, αij z j , Qi (z) := 1 ≤ i ≤ N. j=0 Each of them is non-zero and satisﬁes Qi (A)ei = 0. Thus, the product polynomial Q := Q1 · · · QN satisﬁes Q = 0 and Q(A)ei = 0, 1 ≤ i ≤ N, since Q1 (A), .