Read e-book online Algebraic Multiplicity of Eigenvalues of Linear Operators PDF

By Julián López-Gómez

ISBN-10: 376438400X

ISBN-13: 9783764384005

This publication brings jointly all on hand effects in regards to the concept of algebraic multiplicities, from the main vintage effects, just like the Jordan Theorem, to the latest advancements, just like the specialty theorem and the development of the multiplicity for non-analytic households. half I (first 3 chapters) is a vintage direction on finite-dimensional spectral conception, half II (the subsequent 8 chapters) provides the main basic effects to be had in regards to the lifestyles and area of expertise of algebraic multiplicities for genuine 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 compatible for college students on the complicated undergraduate or starting graduate level.

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Let λ1 , . . , λN be the eigenvalues of A, where each eigenvalue is repeated according to its multiplicity. Then, N N λj tr A = j=1 and det A = λj . j=1 34 11. Chapter 1. The Jordan Theorem Let A ∈ MN (R) be a diagonal matrix such that (A + I)2 = 0. Determine its Jordan canonical form. 12. Determine all matrices A ∈ M3 (C) satisfying the equation A3 − 2A2 + A = 0. 13. Let A ∈ MN (C) be such that σ(A) = {α, β}, ν(α) = 2, ν(β) = 1. Construct a polynomial P such that P (A) = 0. 14. Let A ∈ MN (C) be a Hermitian matrix (see Exercise 3).

3 all have real components if λj ∈ R, whereas if λj = αj + iβj , with αj , βj ∈ R, βj = 0, then, their components are complex. Suppose this is the case, and reorder the eigenvalues, if necessary, so that ¯j . 30) where e¯ is the complex conjugate vector of e, that is, the vector whose components are the complex conjugate components of e. Therefore, for each k ≥ 0, dim N [(A − λj I)k ] = dim N [(A − λj+1 I)k ] and, in particular, ν(λj ) = ν(λj+1 ) and ma (λj ) = ma (λj+1 ). Let Bj := {ej1 , . .

Therefore, with respect to the basis C C C C BC := {e1 , e2 , e3 , e4 }, 30 Chapter 1. The Jordan Theorem the associated matrix to A equals its complex Jordan form ⎞ ⎛ 2−i 0 0 0 ⎜ 1 2−i 0 0 ⎟ ⎟ ⎜ JC := ⎜ ⎟ = PC−1 APC ⎝ 0 0 2+i 0 ⎠ 0 0 1 2+i where PC is the matrix of change of basis ⎛ 2 −1 − i ⎜ 0 −1 − i ⎜ PC := ⎜ ⎝ 2i −2i 1 + i −1 − i ⎞ 2 −1 + i 0 −1 + i⎟ ⎟ ⎟. 2 The real Jordan form of A The real Jordan form is obtained by considering the new basis R R R R BR := {e1 , e2 , e3 , e4 }, where ⎛ ⎞ 2 ⎜ ⎟ R C ⎜0⎟ e1 := Re e1 = ⎜ ⎟ , ⎝0⎠ 1 ⎛ ⎞ −1 ⎜ ⎟ R C ⎜−1⎟ e3 := Re e2 = ⎜ ⎟ , ⎝0⎠ −1 Since R Ae1 R Ae2 R Ae3 R Ae4 ⎛ ⎞ 0 ⎜ ⎟ R C ⎜0⎟ e2 := Im e1 = ⎜ ⎟ , ⎝2⎠ 1 ⎛ ⎞ −1 ⎜ ⎟ R C ⎜−1⎟ e4 := Im e2 = ⎜ ⎟ .

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Algebraic Multiplicity of Eigenvalues of Linear Operators (Operator Theory: Advances and Applications) by Julián López-Gómez


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