Numerical range of a linear transformation

#linear_algebra

Suppose $T$ is a linear transformation on a finite-dimensional complex inner-product space $V$ . Let $I$ denote the identity transformation on $V$ . The numerical range of $T$ is the subset of $C$ defined by

W (T) = {⟨ T (x), x ⟩ | x \in V, ∥ x ∥ = 1} .

Show that $W (T + c I) = W (T) + c$ for every $c \in C$ .
Show that $W (c T) = c W (T)$ for every $c \in C$ .
Show that the eigenvalues of $T$ are contained in $W (T)$ .
Let $B$ be an orthonormal basis for $V$ . Show that the diagonal entries of $[T]_{B}$ are contained in $W (T)$ .

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L A T E X

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Suppose $T$ is a linear transformation on a finite-dimensional complex inner-product space $V$. Let $I$ denote the identity transformation on $V$. The {\bfseries numerical range} of $T$ is the subset of ${\bf C}$ defined by
\[
	\operatorname{W}(T)=\{\langle T(x),x\rangle \,|\, x\in V,\quad \|x\|=1\}.
\]

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Show that $\operatorname{W}(T+cI)=\operatorname{W}(T)+c$ for every $c\in {\bf C}$.
	\item Show that $\operatorname{W}(cT)=c\operatorname{W}(T)$ for every $c\in {\bf C}$.
	\item Show that the eigenvalues of $T$ are contained in $\operatorname{W}(T)$.
	\item Let $\mathcal{B}$ be an orthonormal basis for $V$. Show that the diagonal entries of $[T]_{\mathcal{B}}$ are contained in $\operatorname{W}(T)$.
\end{enumerate}

Important note

Sometime after this exam was given, the exam syllabus was updated and the topic of general inner-product spaces was removed. As such, this problem does not appear in the problem bank.