With each A RnÃn we can associate n closed circular disks in the complex plan. The ith

Question:

With each A ˆˆ RnÃ—n we can associate n closed circular disks in the complex plan. The ith disk is centered at aij and has radius

Each eigenvalue of A is contained in at least one of the disks.
If k of the Gerschgorin disks form a connected domain in the complex plane that is isolated from the other disks, then exactly k of the eigenvalues of the matrix will lie in that domain. Set
B = [3 0.1 2; 0.1 7 2: 2 2 50]
(i) Use the method described in part (a) to compute and plot the Gerschgorin disks of B.
(ii) Since B is symmetric, its eigenvalues are all real and so must all lie on the real axis. Without computing the eigenvalues, explain why B must have exactly one eigenvalue in the interval [46, 54]. Multiply the first two rows of B by 0.1 and then multiply the first two columns by 10. We can do this in MATLAB by setting
D = diag[0.l, 0.1, 11) and C = D*B/D
The new matrix C should have the same eigenvalues as B. Why? Explain. Use C to find intervals containing the other two eigenvalues. Compute and plot the Gerschgorin disks for C.
(iii) How are the eigenvalues of CT related to the eigenvalues of B and C? Compute and plot the Gerschgorin disks for CT. Use one of the rows of CT to find an interval containing its largest eigenvalue.