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Problem 1 Consider the matrix $A = \begin{pmatrix} 0 & 0 & -1 + i \\ 0 & 3 & 0 \\ -1 - i & 0 & 0 \end{pmatrix}$. (a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by $|a_1\rangle$, $|a_2\rangle$, $|a_3\rangle$. Any degenerate eigenvalues? (b) Show that the eigenvectors $|a_1\rangle$, $|a_2\rangle$, $|a_3\rangle$ form an orthonormal and complete basis, i.e., show that $\sum_{j=1}^3 |a_j\rangle\langle a_j| = I$, where I is the $3 \times 3$ unit matrix, and that $\langle a_j | a_k \rangle = \delta_{jk}$. (c) Find the matrix corresponding to the operator obtained from the ket-bra product of the first eigenvector $P = |a_1\rangle\langle a_1|$. Is P a projection operator?

          Problem 1
Consider the matrix $A = \begin{pmatrix} 0 & 0 & -1 + i \\ 0 & 3 & 0 \\ -1 - i & 0 & 0 \end{pmatrix}$.
(a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by $|a_1\rangle$, $|a_2\rangle$, $|a_3\rangle$. Any degenerate eigenvalues?
(b) Show that the eigenvectors $|a_1\rangle$, $|a_2\rangle$, $|a_3\rangle$ form an orthonormal and complete basis, i.e., show that $\sum_{j=1}^3 |a_j\rangle\langle a_j| = I$, where I is the $3 \times 3$ unit matrix, and that $\langle a_j | a_k \rangle = \delta_{jk}$.
(c) Find the matrix corresponding to the operator obtained from the ket-bra product of the first eigenvector $P = |a_1\rangle\langle a_1|$. Is P a projection operator?

Problem 1
Consider the matrix A =
< p m a t r i x >.
(a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by |a1⟩, |a2⟩, |a3⟩. Any degenerate eigenvalues?
(b) Show that the eigenvectors |a1⟩, |a2⟩, |a3⟩ form an orthonormal and complete basis, i.e., show that ∑j=1^3 |aj⟩⟨ aj| = I, where I is the 3 × 3 unit matrix, and that ⟨ aj | ak ⟩ = δjk.
(c) Find the matrix corresponding to the operator obtained from the ket-bra product of the first eigenvector P = |a1⟩⟨ a1|. Is P a projection operator?

Added by Jennifer M.

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