5. A 3 x 3 matrix is given by A = 0 20 -i 01 (a) Verify that A is hermitian. (b) Calculate Tr (A) and det (A), where det (A) represents the determinant of A. (c) Find the eigenvalues of A. Check that their product and sum are consistent with Prob. (5b). (d) Write down the diagonalized version of A. (e) Find the three orthonormal eigenvectors of A. (f) Construct the unitary matrix U that diagonalizes A, and show explicitly that the similarity transformation using U reduces A to the appropriate diagonal form. Did this similarity transformation yield the same diagonalized matrix you wrote down in Prob. (5d). If not, why not?
5. A 3 x 3 matrix is given by A = 0 20 -i 01 (a) Verify that A is hermitian. (b) Calculate Tr (A) and det (A), where det (A) represents the determinant of A. (c) Find the eigenvalues of A. Check that their product and sum are consistent with Prob. (5b). (d) Write down the diagonalized version of A. (e) Find the three orthonormal eigenvectors of A. (f) Construct the unitary matrix U that diagonalizes A, and show explicitly that the similarity transformation using U reduces A to the appropriate diagonal form. Did this similarity transformation yield the same diagonalized matrix you wrote down in Prob. (5d). If not, why not?
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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Just part f) ple
Transcribed Image Text:5. A 3 x 3 matrix is given by
A =
1
020
-i01
(a) Verify that A is hermitian.
(b) Calculate Tr (A) and det (A), where det (A) represents the determinant of A.
(c) Find the eigenvalues of A. Check that their product and sum are consistent with Prob. (5b).
(d) Write down the diagonalized version of A.
(e) Find the three orthonormal eigenvectors of A.
(f) Construct the unitary matrix U that diagonalizes A, and show explicitly that the similarity
transformation using U reduces A to the appropriate diagonal form. Did this similarity
transformation yield the same diagonalized matrix you wrote down in Prob. (5d). If not,
why not?
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VIEWStep 5: (d) Calculating the diagonalised version of A:
VIEWStep 6: (e) Finding the orthogonal vectors of A:
VIEWStep 7: (f) Finding the unitary matrix U that diagonalises A:
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