Math 202C: May 27 Lecture

*** Problems in this lecture due May 31 at 23:59 ***

Problem 1. Let $V$ be a one-dimensional Hilbert space and $A \in \mathrm{End}(V)$ a linear operator. Prove that there is $\alpha \in \mathbb{C}$ such that $Av = \alpha v$ for all $v \in V$ .

Yes, Problem 1 is trivial. Yes, I want you to write down a proof.

Problem 2. Let $V$ be a Hilbert space of finite dimension $n \in\mathbb{N}$ and let $A \in \mathrm{End}(V)$ be a linear operator. Prove that the function of a complex variable $z$ defined by $P_A(z) =\det(zI-A),$ where $I \in \mathrm{End}(A)$ is the identity operator, can also be expressed as

$P_A(z) = \sum\limits_{d=0}^n z^{n-d} (-1)^d \mathrm{Tr}(A^{\wedge d}).$

Yes, the power of $-1$ is correct.

Problem 3. Prove that the following two statements are equivalent: every linear operator on a finite-dimensional Hilbert space has nonempty spectrum; every polynomial function of a complex variable has a zero.

Problem 4. Let $V$ be a finite-dimensional Hilbert space with orthonormal basis $e_1,\dots,e_n.$ For $1 \leq d \leq n$ and $I \subseteq \{1,\dots,n\}$ a subset with $|I|=d,$ let $V_I$ be the subspace of $V$ spanned by $\{e_i \colon i \in I\}$ and let $A_I$ be the restriction of $A$ to $V_I.$ Prove that

$\mathrm{Tr}(A^{\wedge d}) = \sum\limits_{\substack{I \subseteq \{1,\dots,n\} \\ |I|=d}} \det(A_I).$

Problem 5. With notation as in Problem 2, write $P_A(z)=(z-\alpha_1)\dots (z-\alpha_n).$ Determine the eigenvalues of $A^{\otimes d},$ $A^{\wedge d},$ and $A^{\vee d}$ in terms of $\alpha_1,\dots,\alpha_n.$

Math 202C: May 27 Lecture

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