Appendix B — Homework 1

Exercise B.1 Let \(P: V \to V\) be a projection.

  1. Show that \(\mathbb{I}-P: V \to V\) is also a projection, that \(\ker P = \operatorname{im}(\mathbb{I}-P)\), and that \(V = \ker P \oplus \operatorname{im}P\).

  2. Show that an eigenvalue of a projection \(P\) can only be \(0\) or \(1\). Conclude that \(\operatorname{Tr}(P) = \dim(\operatorname{im}P)\), i.e., the trace computes the dimension of the subspace projected onto by \(P\).

Exercise B.2 If \(V\) is an inner product space and \(\|x_0\| = 1\), show \(P(x) \coloneqq \langle x_0, x \rangle x_0\) is an orthogonal projection (cf. Remark 2.6) onto \(\operatorname{Span}\{x_0\}\).

Exercise B.3 Show that a pairwise orthonormal set \(\mathscr{A}\) of an inner product space \(V\) is necessarily a linearly independent set.

Exercise B.4 Verify that the set \(S = \left\{ \frac{1}{\sqrt{2\pi}} e^{int}\ \middle|\ n \in \mathbb{Z}\right\}\) is pairwise orthonormal in the inner product space \(C([-\pi,\pi],\mathbb{C})\) with respect to \[ \langle f, g \rangle \coloneqq \int_{-\pi}^{\pi} \overline{f(t)} g(t) \mathop{}\!\mathrm{d}{t}. \]

Exercise B.5 Let \(Q \in \operatorname{Mat}_n(\mathbb{C})\). Prove that the following are equivalent:

  1. \(Q \in \operatorname{U}(n)\) (see Definition 2.29).
  2. The columns of \(Q\) are orthonormal with respect to the Hermitian dot product.
  3. The rows of \(Q\) are orthonormal with respect to the Hermitian dot product.

Exercise B.6 Let \(A = \begin{psmallmatrix} -10 & 18 & 18 \\ 3 & -7 & -6 \\ -9 & 18 & 17 \\ \end{psmallmatrix}\). Compute \(A^{10}\).

Remark: You can easily do this by brute force with a computer algebra system like SageMath, but that isn’t the point—we can do this manually! The hint here is: if \(S^{-1} A S\) is diagonal, computing its powers is easy. Does that help with powers of \(A\)?

Exercise B.7 Let \(V\) be an inner product space with \(L: V \to V\) a linear transformation.

  1. Show that \(L\) can be decomposed as \(L = A+Bi\), where \(A\) and \(B\) are self-adjoint operators.

  2. Show that \([L^*,L] = 2i[A,B],\) i.e., \(L\) is normal if and only if its real and imaginary part commute.

  3. (Bonus) Suppose we have proven the spectral theorem (Theorem 2.19) for Hermitian matrices: Every \(M\) satisfying \(M = M^*\) admits some \(Q \in \operatorname{U}(n)\) such that \(Q^* M Q\) is diagonal. Using this, prove the spectral theorem for normal matrices.

    Hint: Use (b) and modify the proof of Theorem 2.15 to show commuting normal matrices admit a simultaneous unitary diagonalization.

  4. Show that the sum and product of commuting normal matrices is normal.

    Remark: You may use the content and results of part (c) even if you did not prove them: Theorem 2.19 and the fact that, when two normal matrices commute, they simultaneously unitarily diagonalize.