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Exercise 58. Verify that for P = then PAP-¹ = (+9)

Exercise 58. Verify that for P = then PAP-¹ = (+9)

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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Exercise 58. Verify that for P = Let: 1 -i Is there a geometric interpretation for the action of A with respect to this basis? Note that for v = (a, b)' = R², Av = (b, –a)' has the same Euclidean length as v. Moreover, the angle between v and Av is always 90°. (One can deduce the latter claim from the humble result from Analytic Geometry that says lines having slopes with a product of -1 are perpendicular lines.) A then PAP-¹ = Hence A can be thought of as a counterclockwise rotation by 90°. We can generalize this result a bit: = (-₁5) (18) (19)
Transcribed Image Text:Exercise 58. Verify that for P = Let: 1 -i Is there a geometric interpretation for the action of A with respect to this basis? Note that for v = (a, b)' = R², Av = (b, –a)' has the same Euclidean length as v. Moreover, the angle between v and Av is always 90°. (One can deduce the latter claim from the humble result from Analytic Geometry that says lines having slopes with a product of -1 are perpendicular lines.) A then PAP-¹ = Hence A can be thought of as a counterclockwise rotation by 90°. We can generalize this result a bit: = (-₁5) (18) (19)
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