MATRIX THEORY CHAPTER 6 FALL 2017 1.GERSGORIN DISK THEOREM Theorem 1 (Geirsgorin).Let A=[alE Mn and let R,=∑lal,1≤i≤n. ji Then (1)all the eigenvalues of A are located in the union of the n-discs UzeC:-al≤R}=G(A) (2)if a union of k dises forms a comnected region that is disjoint from all the remaining n-k discs. then there are eract k eigenvalues of A in this region. Corollary 1.Let A=M and let C=∑lal1≤j≤n 南 Then (1)all the eigenvalues of A are located in the union of the n-discs U[zeC:2-al≤C}=G(AT方 j=1 (2)if a union of k discs forms a connected region that is disjoint from all the remaining n-k discs, then there are eract k eigenvalues of A in this region. Corollary 2.If A=[]Mn,then p(4≤min{max∑lal.max∑al}=min{lAle,lA} 1■1 Corollary 3.Let A=a]Mn and let D=diag(pi,pa.....),where p,pa.pn are positive real numbers.Then all the eigenvalues of A lie in the region tec:-s-(o-tAD) as well as in the region U(EC:k-apls>plaull -G(D-ATD). 卫街 Example 1.Let 21-4 MATRIX THEORY - CHAPTER 6 FALL 2017 1. Gersgorin Disk Theorem Theorem 1 (Geirsgorin). Let A = [aij ] ∈ Mn and let Ri = X j6=i |aij |, 1 ≤ i ≤ n. Then (1) all the eigenvalues of A are located in the union of the n-discs [n i=1 {z ∈ C : |z − aii| ≤ Ri} = G(A); (2) if a union of k discs forms a connected region that is disjoint from all the remaining n − k discs, then there are exact k eigenvalues of A in this region. Corollary 1. Let A = [aij ] ∈ Mn and let Cj = X i6=j |aij |, 1 ≤ j ≤ n. Then (1) all the eigenvalues of A are located in the union of the n-discs [n j=1 {z ∈ C : |z − ajj | ≤ Cj} = G(A T ); (2) if a union of k discs forms a connected region that is disjoint from all the remaining n − k discs, then there are exact k eigenvalues of A in this region. Corollary 2. If A = [aij ] ∈ Mn, then ρ(A) ≤ min{max i Xn j=1 |aij |, max j Xn i=1 |aij |} = min{9A9∞,9A91} Corollary 3. Let A = [aij ] ∈ Mn and let D = diag{p1, p2, ..., pn}, where p1, p2, · · · pn are positive real numbers.Then all the eigenvalues of A lie in the region [n i=1 {z ∈ C : |z − aii| ≤ 1 pi X j6=i pj |aij |} = G(D−1AD) as well as in the region [n j=1 {z ∈ C : |z − ajj | ≤ 1 pj X i6=j pi |aij |} = G(D−1A T D). Example 1. Let A =   2 0 1 1 3 1 2 1 −4   Corollary 4. Let A be a real square matrix. Comsider one connected component of G(A) (or G(AT )), which is a union of k-disks. If k is odd, then in this component there is at least one real eigenvalue. 1