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Q3. Consider the matrix, A = UT AU 0 0 0 1 0 -1 0 0 (a) Find the eigenvalues A's (doubly degenerate) and the eigenvectors ([la,)},i=1,2,3)= the operator A. (b) Show that each of the sets la₁), la₂), la3) forms an orthonormal and complete basis. (c) Consider the matrix U which is formed from the normalized eigenvectors of A. Verify that U is unitary (U*U = 1). (d) Show that U satisfies the relation = 12₁ 0 0 0 ₂ 0 0 0, A₁, A₂, A3 are the eigenvalues of A. A3

Q3. Consider the matrix, A = UT AU 0 0 0 1 0 -1 0 0 (a) Find the eigenvalues A's (doubly degenerate) and the eigenvectors ([la,)},i=1,2,3)= the operator A. (b) Show that each of the sets la₁), la₂), la3) forms an orthonormal and complete basis. (c) Consider the matrix U which is formed from the normalized eigenvectors of A. Verify that U is unitary (U*U = 1). (d) Show that U satisfies the relation = 12₁ 0 0 0 ₂ 0 0 0, A₁, A₂, A3 are the eigenvalues of A. A3

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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Q3. Consider the matrix, A = 0 0 -1 UT AU= = 0 -11 12₁₂ 0 0 1 0 Foo 0 (a) Find the eigenvalues A's (doubly degenerate) and the eigenvectors ([la)},i= 1,2,3) of the operator A. (b) Show that each of the sets la₁), la₂), la3) forms an orthonormal and complete basis. (c) Consider the matrix U which is formed from the normalized eigenvectors of A. Verify that U is unitary (U*U = 1). (d) Show that U satisfies the relation 0 0 0 1₂0₁, A2, A3 are the eigenvalues of A. 0 234
Transcribed Image Text:Q3. Consider the matrix, A = 0 0 -1 UT AU= = 0 -11 12₁₂ 0 0 1 0 Foo 0 (a) Find the eigenvalues A's (doubly degenerate) and the eigenvectors ([la)},i= 1,2,3) of the operator A. (b) Show that each of the sets la₁), la₂), la3) forms an orthonormal and complete basis. (c) Consider the matrix U which is formed from the normalized eigenvectors of A. Verify that U is unitary (U*U = 1). (d) Show that U satisfies the relation 0 0 0 1₂0₁, A2, A3 are the eigenvalues of A. 0 234
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