(b) By considering the Gerschgorin circles, show that the eigenvalues of the matrix 2 -1 0 A = -1 2-1 0-1 2 satisfy the inequality 0 = λ = 4 Hence, using the result proved in (a), determine the smallest modulus eigenvalue of A correct to two decimal places.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.3: The Gram-schmidt Process And The Qr Factorization
Problem 8BEXP
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1.5.3 Only B
(a) If the eigenvalues of the n x n matrix A are
λ > λ > 13 ... λ = 0
n
show that the eigenvalue A, can be found by
applying the power method to the matrix kl - A,
where is the identity matrix and k> 2₁.
(b) By considering the Gerschgorin circles, show
that the eigenvalues of the matrix
2-1 0
A =-1 2-1
0-1 2
satisfy the inequality
0≤2≤4
Hence, using the result proved in (a), determine
the smallest modulus eigenvalue of A correct to
two decimal places.
Transcribed Image Text:1.5.3 Only B (a) If the eigenvalues of the n x n matrix A are λ > λ > 13 ... λ = 0 n show that the eigenvalue A, can be found by applying the power method to the matrix kl - A, where is the identity matrix and k> 2₁. (b) By considering the Gerschgorin circles, show that the eigenvalues of the matrix 2-1 0 A =-1 2-1 0-1 2 satisfy the inequality 0≤2≤4 Hence, using the result proved in (a), determine the smallest modulus eigenvalue of A correct to two decimal places.
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