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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcea2167e-7bcc-4d90-9a35-3526a4009c32%2Fc10e3226-a28c-41f8-9737-af41c495c378%2Fwdwrnk_processed.png&w=3840&q=75)
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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