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Consider the matrix A = 0.6 -0.1 0.1 0.1 0.1 0.6 -0.1 -0.1 0.1 0.1 -0.6 0.1 -0.15 0.15 0.15 0.5 (a) Use Gershgorin's Circle theorem to find an upper bound on the spectral radius p(A). (b) What can you say about lim Ak given your upper bound on the spectral radius of A? k→∞ (c) Prove that A has exactly one negative real eigenvalue. Hint: Draw the Gershgorin disks in the complex plane and use the second part of the full version of the Gershgorin Circle Theorem from module 3.

Consider the matrix A = 0.6 -0.1 0.1 0.1 0.1 0.6 -0.1 -0.1 0.1 0.1 -0.6 0.1 -0.15 0.15 0.15 0.5 (a) Use Gershgorin's Circle theorem to find an upper bound on the spectral radius p(A). (b) What can you say about lim Ak given your upper bound on the spectral radius of A? k→∞ (c) Prove that A has exactly one negative real eigenvalue. Hint: Draw the Gershgorin disks in the complex plane and use the second part of the full version of the Gershgorin Circle Theorem from module 3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the matrix A = 0.6 0.1 0.1 0.1 -0.1 0.6 -0.1 -0.1 0.1 0.1 -0.6 0.1 -0.15 0.15 0.15 0.5 (a) Use Gershgorin's Circle theorem to find an upper bound on the spectral radius p(A). (b) What can you say about lim Ak given your upper bound on the spectral radius of A? k→∞0 (c) Prove that A has exactly one negative real eigenvalue. Hint: Draw the Gershgorin disks in the complex plane and use the second part of the full version of the Gershgorin Circle Theorem from module 3.
Transcribed Image Text:Consider the matrix A = 0.6 0.1 0.1 0.1 -0.1 0.6 -0.1 -0.1 0.1 0.1 -0.6 0.1 -0.15 0.15 0.15 0.5 (a) Use Gershgorin's Circle theorem to find an upper bound on the spectral radius p(A). (b) What can you say about lim Ak given your upper bound on the spectral radius of A? k→∞0 (c) Prove that A has exactly one negative real eigenvalue. Hint: Draw the Gershgorin disks in the complex plane and use the second part of the full version of the Gershgorin Circle Theorem from module 3.
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