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By Kalnin, Robert Avgustovich

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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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Algebra y funciones elementales by Kalnin, Robert Avgustovich


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