Up to this point, our main theoretical tools for finding eigenvalues without using det{a λi} = 0 have been the trace and determinant formulas


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1 Finding Eigenvalues Up to this point, our main theoretical tools for finding eigenvalues without using det{a λi} = 0 have been the trace and determinant formulas plus the facts that det{a} = λ λ λ n, Tr{A} = λ +λ + +λ n the eigenvalues of a triangular matrix are the diagonal elements similar matrices B = S AS have the same eigenvalues the eigenvalues of a real symmetric matrix are real the eigenvalues of an orthogonal matrix have λ =. The first fact can be generalized to block triangular matrices. If M is block upper triangular [ A B M = 0 C the eigenvalues of M are the eigenvalues of A plus the eigenvalues of C since [ A λi B 0 = det{m λi} = det = det{a λi}det{c λi}. 0 C λi Here each identity matrix I has the appropriate dimensions to match its partner. A similar statement holds for block lower triangular M. Thus for block triangular matrices, the eigenvalue problem can be broken up into smaller subproblems. (A matrix M is reducible if it can be written in block triangular form by reorderingrowsandcolumnspmp, wherep isapermutationmatrix.) Gershgorin Circle Theorem Also called the Gershgorin Disk Theorem. Theorem statement and Examples and are based on LeVeque s Finite Difference Methods for Ordinary & Partial Differential Equations.
2 Theorem. Let A C n n and let D i be the closed disk in the complex plane centered at A ii with radius given by the row sum r i = j i A ij : D i = {z C : z A ii r i } D(A ii,r = r i ). ThenalltheeigenvaluesofAlieintheunionofthedisksD i fori =,..., n. If some set of k overlapping disks is disjoint from all the other disks, then exactly k eigenvalues lie in the union of these k disks. Note the following: If a disk D i is disjoint from all other disks, then it contains exactly one eigenvalue of A. If a disk D i overlaps other disks, then it need not contain any eigenvalues, although the union of the overlapping disks contains the appropriate number. If A is real, A T has the same eigenvalues as A and the theorem can also be applied to A T (or equivalently the disk radii can be defined by summing the magnitudes of the offdiagonal elements of columns of A rather than rows). (If A is irreducible, a stronger version of the theorem states that an eigenvalue cannot lie on the boundary of a disk unless it lies on the boundary of every disk.) Proof: Ax = λx implies (λ A ii )x i = j i A ij x j λ A ii j i by choosing i such that x i = max j x j. A ij x j x i A ij = r i j i
3 Examples Here we assume all matrices are real example A = The Gershgorin theorem applied to A implies that the eigenvalues lie within the union of D(5,r = 0.7), D(6,r =.), and D(,r = ). There is exactly one eigenvalue in D(,r = ) and two eigenvalues in D(5,r = 0.7) D(6,r =.). All eigenvalues have real parts between and 7. (and hence positive real parts, in particular). The eigenvalue in D(, r = ) must be real, since complex eigenvalues must appear in conjugate pairs. Applying the theorem to A T gives a tighter bound on the single eigenvalue λ near ; λ must lie within D(,r = 0.), so.8 λ.. The actual eigenvalues of A are λ =.9639 and λ ± = 5.580±0.64i.. Second difference matrix ThematrixD () = tridiag[ issymmetric, soallitseigenvaluesarereal. By the Gershgorin theorem they must lie in the circle of radius centered at : 4 λ j 0. We know that D () is nonsingular, so all the eigenvalues of D () are negative. (Also since D () is irreducible and two circles have radii =, 4 < λ j < 0, which proves D () is nonsingular.) The eigenvalues can actually be calculated: λ j = (cos(jπh) ), j =,..., n where h = /(n+), and thus are distributed between 4 < λ j < 0. The Jacobi iteration matrix is B = tridiag[ 0 with < λ j < (since B is irreducible), and Jacobi converges since ρ(b) <. 3. Absolute row sums < Suppose B has absolute row sums <. Then r i = B i + + B i, + B i+, + + B in < B ii 3
4 and by the Gershgorin theorem all λ i < and ρ(b) <. 4. Diagonally dominant matrix Suppose A is diagonally dominant: r i = A i + + A i, + A i+, + + A in < A ii, i =,..., n. Then by the Gershgorin theorem λ i 0 and A is invertible. For Jacobi iteration, B = I D A has all absolute row sums <, so Jacobi converges. GaussSeidel and SOR are also guaranteed to converge. In Examples 5 9, see how much information you can extract about the actual eigenvalues using just the Gershgorin Circle Theorem and (block) triangularity. The actual eigenvalues are given. 5. From HW6 λ = 0, λ ± = ± /. 6. From HW7 λ ± = (± 5)/. A = A = [ 0 Last examples are from Theory of Iterative Methods notes 7. For B GS = B J = [ A = [ 0 [ , λ ± = ±, ρ J =, λ = 0, λ = 4, ρ GS = 4 8. Example where Jacobi converges but GaussSeidel diverges 0 A =, B J = 0 0 4
5 λ = λ = λ 3 = 0, ρ J = 0, Jacobi converges. 0 B GS = λ = 0, λ = λ 3 =, ρ GS =, GaussSeidel diverges. 9. Example where Jacobi diverges but GaussSeidel converges A =, B J = λ =, λ = λ 3 = 0.5, ρ J =, Jacobi diverges. B GS = λ = 0, λ ± = 0.35±0.654i, ρ GS = , GaussSeidel converges. 5
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