Consider the following matrix: -2 -2 A =-14 15 7. 18 -18 -10 (a) Find the eigenvalues of A. (b) Find three linearly independent eigenvectors for A and state the algebraic and geometric multiplicity for each eigenvalue. (c) Form a matrix C whose columns are the three linearly independent eigenvectors that you found in (b). Verify that C-AC is a diagonal matrix (which we call D). 3.
Consider the following matrix: -2 -2 A =-14 15 7. 18 -18 -10 (a) Find the eigenvalues of A. (b) Find three linearly independent eigenvectors for A and state the algebraic and geometric multiplicity for each eigenvalue. (c) Form a matrix C whose columns are the three linearly independent eigenvectors that you found in (b). Verify that C-AC is a diagonal matrix (which we call D). 3.
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:4. Consider the following matrix:
3.
-2
A =-14
15
18
-18 -10
(a) Find the eigenvalues of A.
(b) Find three linearly independent eigenvectors for A and state the algebraic and geometric
multiplicity for each eigenvalue.
Form a matrix C whose columns are the three linearly independent eigenvectors that you
found in (b). Verify that C-1 AC is a diagonal matrix (which we call D).
(d) Use your previous answer to find a matrix B with the property that B3 = A.
(e) For any real matrix A, write down a sufficient condition for a real matrix E to exist such
that E2 = A, and prove that this condition is sufficient.
%3D
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