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

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