Algorithmic Linear Algebra by Herbert Möller

By Herbert Möller

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In addition, σ(A− ) = σ(D). 3) complete the proof in the case µ2 (A) ≤ 1. 6 Lower Bounds for the Spectral Radius Let A = (ajk ) be an n × n-matrix. Recall that rs (A) = sup |σ(A)| is the (upper) spectral radius; α(A) = sup Re σ(A) and rl (A) = inf |σ(A)| is the inner (lower) spectral radius. 1). 1) k=0 where ν(A) = min{ V− 2 , V+ 2 }. 1 Let A = (ajk )nj,k=1 be an n × n-matrix. , n, there is an eigenvalue µ0 of A, such that |µ0 − akk | ≤ z(ν). 4) and α(A) ≥ max Re akk − z(ν). 5) 58 4. Eigenvalues of Finite Matrices Proof: Take A+ = D + V+ .

9) ˜k ∞ = w It is not hard to check that V− ∆Q ˜n−k+1 . 9). 10) k=1 are valid. 11) for any nonnegative h ∈ Cn . , n). ,n ∞ − 1. 12) k=1 Here δjk is the Kronecker symbol. Furthermore, since V+ is nilpotent, n−1 (−1)k V+k k=1 n−1 ∞ ≤ |V+ |k ∞ = k=1 (I − |V+ |)−1 − I ∞ where |V+ | is the matrix whose entries are the absolute values of the entries of V . Moreover, clearly, n−1 k=0 |V+ |k h ≥ h 44 3. Invertibility of Finite Matrices for any nonnegative h ∈ Cn . 10). 10) can be proved. 2, (I + W− )−1 ∞ ≤ mup (A), (I + W+ )−1 ∞ ≤ mlow (A).

This yields Rµ (A) − Rµ (B) ≤ Rµ (B) q Rµ (A) . 2) implies Rµ (B) ≤ Rµ (A) (1 − q Rµ (A) )−1 . That is, µ is a regular point of B. This contradiction proves the result. 4) where φ(x) is a monotonically increasing non-negative function of a nonnegative variable x, such that φ(0) = 0 and φ(∞) = ∞. 5) is true, where z(φ, q) is the extreme right-hand (positive) root of the equation 1 = qφ(1/z). 4), 1 ≤ qφ(ρ−1 (A, λ)) for all λ ∈ σ(B). 6) and ρ(A, λ) ≤ z(φ, q). 5) is valid. 2 Perturbations of Multiplicities and Matching Distance Put Ω(c, r) ≡ {z ∈ C : |z − c| ≤ r} (c ∈ C, r > 0).

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