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(a) Let A = 5 3 -3 1 1] 3 1 -1 1 " which has characteristic polynomial C₁(x) = -(x - 2)²(x - 5). (i) From the polynomial C₁ (2) alone, what can one say about the geometric multiplicities of the eigenvalues 2 and 5? Be as specific as you can. (ii) Now find a basis for the eigenspace associated to the eigenvalue 2, and hence give the geometric multiplicity of that eigenvalue.

(a) Let A = 5 3 -3 1 1] 3 1 -1 1 " which has characteristic polynomial C₁(x) = -(x - 2)²(x - 5). (i) From the polynomial C₁ (2) alone, what can one say about the geometric multiplicities of the eigenvalues 2 and 5? Be as specific as you can. (ii) Now find a basis for the eigenspace associated to the eigenvalue 2, and hence give the geometric multiplicity of that eigenvalue.

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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(a) Let A = LO 30 5 B = 3 -3 1 3 -1 which has characteristic polynomial CÂ(x) = − (x − 2)²(x – 5). (i) From the polynomial CA (2) alone, what can one say about the geometric multiplicities of the eigenvalues 2 and 5? Be as specific as you can. 1 " (ii) Now find a basis for the eigenspace associated to the eigenvalue 2, and hence give the geometric multiplicity of that eigenvalue. (b) Let C³ 1 c² 0 C-3 0 с 0 ³+1 where c E R. How many eigenvalues does B have and with what ge- ometric multiplicities? Without knowing c, is it possible to determine whether B is diagonalizable? Explain your answers.
Transcribed Image Text:(a) Let A = LO 30 5 B = 3 -3 1 3 -1 which has characteristic polynomial CÂ(x) = − (x − 2)²(x – 5). (i) From the polynomial CA (2) alone, what can one say about the geometric multiplicities of the eigenvalues 2 and 5? Be as specific as you can. 1 " (ii) Now find a basis for the eigenspace associated to the eigenvalue 2, and hence give the geometric multiplicity of that eigenvalue. (b) Let C³ 1 c² 0 C-3 0 с 0 ³+1 where c E R. How many eigenvalues does B have and with what ge- ometric multiplicities? Without knowing c, is it possible to determine whether B is diagonalizable? Explain your answers.
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