By Mahdi Alosh

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J=1 Show that each of these norms give rise to the same set Gφ (A; B) in the complex plane, as shown in Fig. 9. (The eigenvalues of A are also shown in Fig. ”) x 2. 57) is a norm on Cn . 3. 25. 4. For the matrices ⎡ ⎤ 1 0 0 A=⎣ 1 5 1 ⎦ 1 1 5 verify that if φ is the ∞ -norm ⎡ ⎤ 1 0 0 and B = ⎣ 0 5 0 ⎦ 0 0 5 on C3 , then 30 1. Basic Theory z ∈C : z∈ / σ(B) and (zI −B)−1 (A−B) φ ≥1 = {z ∈ C : |z−5| ≤ 2}, which does not contain the spectrum, σ(A) = {1, 4, 6}, of A. 48) to insure that Gϕ (A; B) covers the spectrum of A.

48) alone contained all the eigenvalues of A. See Exercise 4 of this section for a counterexample. 22. For any A ∈ Cn×n , and any B ∈ Cn×n , let ϕ be any norm on Cn . Then (cf. 48)), σ(A) ⊆ Gϕ (A; B). 49), is an eigenvalue inclusion set for any matrix A. We now deduce some settheoretic properties of Gϕ (A; B). 23. For any A ∈ Cn×n , and any B ∈ Cn×n let ϕ be any norm on Cn . 48)), Gϕ (A; B) is a closed and bounded set in C. Proof. We ﬁrst show that Gϕ (A; B) is bounded. Suppose that z ∈ Gϕ (A; B) −1 with z ∈ σ(B), so that 1 ≤ (zIn − B) (A − B) ϕ .

33). 33), we describe below a modest extension of an important work of Brualdi (1982), which introduced the notion of a cycle3 , from the directed graph of A, to obtain an eigenvalue inclusion region for any A. This extension is also derived from properties of the directed graph of the matrix A. 2, on n distinct vertices {vi }ni=1 , which consists of a → (directed) arc − v− i vj , from vertex vi to vertex vj , only if ai,j = 0. ) A strong cycle γ in G(A) is deﬁned as a sequence {ij }p+1 j=1 of integers in N such that p ≥ 2, the elements −−−−−→ −→ v− of {ij }pj=1 are all distinct with ip+1 = i1 , and − i1 vi2 , · · · , vip vip+1 are arcs of G(A).