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The Lumenary · Advanced undergraduate

Advanced linear algebra

Name: ______________________

Date: ______________________

You will learn to

Prove the spectral theorem for self-adjoint operators (real symmetric and complex Hermitian matrices) and for complex normal operators, and orthogonally diagonalise symmetric matrices
Prove the Cayley–Hamilton theorem, compute minimal polynomials, and decide diagonalisability from them
Compute Jordan normal forms from the ranks of powers of $A - \lambda I$ , and prove existence and uniqueness of the Jordan form
Define dual spaces and dual maps, and explain why $V \cong V^{**}$ is natural while $V \cong V^*$ depends on a choice
Define the tensor product by its universal property, compute with the basis $e_i \otimes f_j$ , and show that not every tensor is a pure tensor

1.
The symmetric matrix $A = \begin{pmatrix} 5 & 2 \\ 2 & 2 \end{pmatrix}$ is the matrix of $q(x, y) = 5x^2 + 4xy + 2y^2$ . What is the largest eigenvalue of $A$ , that is, the maximum of $q$ on the unit circle?
2.
Which of these matrices are unitarily (or, for real ones, orthogonally) diagonalisable over $\mathbb{C}$ ? (Choose all that apply.)
- A. $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$
- B. $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
- C. $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$
- D. $\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}$
- E. $\begin{pmatrix} 1 & 10 \\ 0 & 2 \end{pmatrix}$
3.
In the proof of the spectral theorem, $v_1$ is an eigenvector of the self-adjoint operator $T$ and $W = v_1^{\perp}$ . Why does $T$ map $W$ into $W$ ?
- A.For $w \in W$ : $\langle v_1, Tw \rangle = \langle Tv_1, w \rangle = \lambda_1 \langle v_1, w \rangle = 0$ .
- B.Because $T$ is invertible, so it maps a hyperplane onto a hyperplane.
- C.Because $W$ is an eigenspace of $T$ .
- D.Because the eigenvalues of $T$ are real.
4.
A nilpotent operator $N$ on a $5$ -dimensional space has $\operatorname{rank} N = 3$ , $\operatorname{rank} N^2 = 1$ and $N^3 = 0$ . What is the degree of its minimal polynomial?
5.
Let $V$ and $W$ be finite-dimensional. Which statements about $V \otimes W$ are true? (Choose all that apply.)
- A. $\dim(V \otimes W) = \dim V \cdot \dim W$ .
- B.Every element is a finite sum of pure tensors $v \otimes w$ .
- C.Every element is a pure tensor $v \otimes w$ .
- D. $v \otimes w = 0$ if and only if $v = 0$ or $w = 0$ .
- E. $(v_1 + v_2) \otimes (w_1 + w_2) = v_1 \otimes w_1 + v_2 \otimes w_2$ .
6.
Let $V = W = \mathbb{R}$ and let $\eta : \mathbb{R} \to \mathbb{R}^*$ be the 'basis' isomorphism sending $1$ to the functional $1^*$ with $1^*(t) = t$ . Let $f : \mathbb{R} \to \mathbb{R}$ be $f(t) = 2t$ . Compute the number $\big(f^*(\eta(f(1)))\big)(1)$ . (Naturality would require it to equal $\eta(1)(1) = 1$ .)
7.
Which Lean statement correctly formalises Lemma 2 (eigenvectors of a symmetric operator for distinct eigenvalues are orthogonal)? In each, T : E →ₗ[ℝ] E on a real inner product space E.
- A.(hT : T.IsSymmetric) (hv : T v = a • v) (hw : T w = b • w) (hab : a ≠ b) : inner ℝ v w = 0
- B.(hT : T.IsSymmetric) (hv : T v = a • v) (hw : T w = b • w) : inner ℝ v w = 0
- C.(hv : T v = a • v) (hw : T w = b • w) (hab : a ≠ b) : inner ℝ v w = 0
- D.(hT : T.IsSymmetric) (h : inner ℝ v w = 0) : ∃ a b, a ≠ b ∧ T v = a • v ∧ T w = b • w
8.
What is $\dim\big(\mathbb{R}^3 \otimes \mathbb{R}^4 \otimes \mathbb{R}^2\big)$ ?
9.
Implement the uniqueness part of Theorem 12. Write block_sizes(ranks) where ranks = [r0, r1, r2, ...] lists $\operatorname{rank}(A - \lambda I)^k$ for $k = 0, 1, 2, \dots$ , ending once the ranks stop changing. Return the list of Jordan block sizes for $\lambda$ , largest first. (Blocks of size $\geq k$ number $r_{k-1} - r_k$ .)
Computational — Practice

Answer key — Advanced linear algebra

1.
$\chi_A(\lambda) = \lambda^2 - 7\lambda + 6 = (\lambda - 6)(\lambda - 1)$ , so the eigenvalues are $6$ and $1$ . By the variational characterisation in Lemma 4, $\max_{\|v\| = 1} q(v) = 6$ , attained at the eigenvector $(2, 1)/\sqrt5$ : indeed $q(2, 1)/5 = (20 + 8 + 2)/5 = 6$ .
2.
By Theorem 6 we need $AA^* = A^*A$ . The symmetric real matrix and the rotation are normal. The shear and the complex symmetric matrix are not diagonalisable at all ( $J_2(1)$ , and a non-zero matrix with $A^2 = 0$ ). The last matrix is diagonalisable but not normal: $AA^{\mathsf T} = \begin{pmatrix} 101 & 20 \\ 20 & 4 \end{pmatrix} \ne \begin{pmatrix} 1 & 10 \\ 10 & 104 \end{pmatrix} = A^{\mathsf T}A$ , so its eigenvectors cannot be orthogonal.
3.
For $w \in W$ , $\langle v_1, Tw \rangle = \langle Tv_1, w \rangle = \langle \lambda_1 v_1, w \rangle = \lambda_1 \langle v_1, w \rangle = 0$ (with $\lambda_1$ real). So $Tw \in W$ . Without self-adjointness this fails: for $\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$ the eigenvector $e_1$ has $e_1^\perp = \mathbb{R} e_2$ , but $Te_2 = (1, 2) \notin \mathbb{R} e_2$ .
4.
Blocks of size $\ge 1$ : $5 - 3 = 2$ ; of size $\ge 2$ : $3 - 1 = 2$ ; of size $\ge 3$ : $1 - 0 = 1$ ; of size $\ge 4$ : $0$ . So the Jordan form is $J_3(0) \oplus J_2(0)$ (sizes add to $5$ ✓). The minimal polynomial is $x^3$ : indeed $N^3 = 0$ but $N^2 \neq 0$ . Degree $3$ .
5.
True: the dimension formula, spanning by pure tensors, and the zero criterion (a rank-one matrix $vw^{\mathsf T}$ vanishes only if a factor does). False: not every tensor is pure (the identity matrix has rank 2), and $\otimes$ is bilinear, not 'additive in both at once': $(v_1 + v_2) \otimes (w_1 + w_2)$ has four terms.
6.
$f(1) = 2$ ; $\eta(2) = \varphi$ with $\varphi(t) = 2t$ ; $f^*(\varphi) = \varphi \circ f$ is $t \mapsto 2 \cdot 2t = 4t$ ; evaluated at $1$ this is $4 \neq 1$ . The scaling $f$ enters twice, once covariantly and once contravariantly, which is the heart of Proposition 15: a natural $\eta$ would satisfy $\eta = c^2\eta$ for every $c \ne 0$ .
7.
The theorem needs symmetry, the two eigen-equations and a ≠ b. Dropping a ≠ b allows w = v; dropping symmetry allows the non-normal matrix of the misconception; the last option reverses the implication. The Lean proof mirrors the lesson: a * inner v w = inner (T v) w = inner v (T w) = b * inner v w.
8.
By Theorem 16 (applied twice), a basis is $e_i \otimes f_j \otimes g_k$ with $1 \le i \le 3$ , $1 \le j \le 4$ , $1 \le k \le 2$ , so the dimension is $3 \cdot 4 \cdot 2 = 24$ .
9.
```
def block_sizes(ranks):
    at_least = [ranks[k - 1] - ranks[k] for k in range(1, len(ranks))]
    at_least.append(0)
    sizes = []
    for k in range(len(at_least) - 1, 0, -1):
        exactly = at_least[k - 1] - at_least[k]
        sizes += [k] * exactly
    return sizes
```
Here at_least[k - 1] is the number of blocks of size at least k. For [5, 3, 1, 0, 0]: at least 1, 2, 3, 4 gives 2, 2, 1, 0, so one block of size 3 and one of size 2. The ranks of $(A - \lambda I)^k$ do not depend on the basis, which is why the Jordan form is unique.