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The eigenvalues of an upper triangular matrix are its diagonal entries, so the matrix has a repeated eigenvalue of a. Check that A" = a"-1 Therefore the effect of A" on vectors is A" (*) a n (₁2) a = an-1 (ax + ny). ay (2.20) (2.21) EXERCISE T2.3 (a) Verify equation (2.20). (b) Use equation (2.21) to show that the fixed point (0, 0) is a sink if |a| < I and a source if lal > 1.

The eigenvalues of an upper triangular matrix are its diagonal entries, so the matrix has a repeated eigenvalue of a. Check that A" = a"-1 Therefore the effect of A" on vectors is A" (*) a n (₁2) a = an-1 (ax + ny). ay (2.20) (2.21) EXERCISE T2.3 (a) Verify equation (2.20). (b) Use equation (2.21) to show that the fixed point (0, 0) is a sink if |a| < I and a source if lal > 1.

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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The eigenvalues of an upper triangular matrix are its diagonal entries, so the matrix has a repeated eigenvalue of a. Check that (aa). n-1 A" = - Therefore the effect of A" on vectors is An ^² ( ² ) = 4²-² (ax + ny) =an-1 ay (2.20) (2.21) EXERCISE T2.3 (a) Verify equation (2.20). (b) Use equation (2.21) to show that the fixed point (0, 0) is a sink if |a| < I and a source if lal > 1.
Transcribed Image Text:The eigenvalues of an upper triangular matrix are its diagonal entries, so the matrix has a repeated eigenvalue of a. Check that (aa). n-1 A" = - Therefore the effect of A" on vectors is An ^² ( ² ) = 4²-² (ax + ny) =an-1 ay (2.20) (2.21) EXERCISE T2.3 (a) Verify equation (2.20). (b) Use equation (2.21) to show that the fixed point (0, 0) is a sink if |a| < I and a source if lal > 1.
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