Q4. Let A = the 2 x 2 identity matrix. (a) Show that the characteristic equation of A is 1² – 21 +1 = 0. Its eigenvalues are A1 = 1 and A2 = 1. (b) We now perturb one coefficient of the characteristic polynomial slightly and con- sider the equation 1² – 21 + (1 – ɛ) = 0, where 0 < ɛ « 1. Solve the equation for the roots A1 and A2. (c) Show that when ɛ = 10-12, JÃ1 – 1| and |^2 – A2| are one million times bigger than ɛ.
Q4. Let A = the 2 x 2 identity matrix. (a) Show that the characteristic equation of A is 1² – 21 +1 = 0. Its eigenvalues are A1 = 1 and A2 = 1. (b) We now perturb one coefficient of the characteristic polynomial slightly and con- sider the equation 1² – 21 + (1 – ɛ) = 0, where 0 < ɛ « 1. Solve the equation for the roots A1 and A2. (c) Show that when ɛ = 10-12, JÃ1 – 1| and |^2 – A2| are one million times bigger than ɛ.
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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Transcribed Image Text:1 0
0 1
Q4. Let A =
the 2 x 2 identity matrix.
(a) Show that the characteristic equation of A is
1? – 21 +1 = 0.
Its eigenvalues are A1 = 1 and A2 = 1.
(b) We now perturb one coefficient of the characteristic polynomial slightly and con-
sider the equation
1² – 21 + (1 – e) = 0,
where 0 < e « 1. Solve the equation for the roots A1 and A2.
(c) Show that when ɛ = 10-12, |X1 – A1| and |X2 – A2| are one million times bigger
than e.
(d) Sketch the graphs of the original and perturbed polynomials (using some ɛ big-
ger than 10-12), to give some indication why the roots are so sensitive to the e
pertubation.
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