Exercise 6.2.9

Question

Give a proof if the statement is true, or give a counterexample if it is false:
\begin{enumerate}[label = \alph*)]
    \item If $A = A^*$ then $A + iI$ is invertible.
    \item If $U$ is unitary, $U+\frac{3}{4}I$ is invertible.
    \item If a matrix is real, $A-iI$ is invertible.
\end{enumerate}

Jing Huang · Accepted Answer

\begin{enumerate}[label = \alph*)]
    \item True. The eigenvalues of $A+iI$ are $\lambda_i + i$ where $\lambda_i$ are eigenvalues of $A$ and are real. Then $\det (A+iI) = \Pi_{i=1}{n} (\lambda_i + i) 
eq 0$. (If $c_1, c_2 \in \mathbb{C}, c_1c_2 = 0$, then at least one of $c_1, c_2$ is 0.)
    \item True. If $(U+\frac{3}{4}I) \mathbf{x} = U\mathbf{x} + \frac{3}{4}\mathbf{x} = \mathbf{0}$, note that $||U\mathbf{x}|| = ||\mathbf{x}||$. Then $||U\mathbf{x} + \frac{3}{4}\mathbf{x}|| \geqslant ||U\mathbf{x}|| - ||\frac{3}{4}\mathbf{x}|| = \frac{1}{4}||\mathbf{x}||$. So the homogeneous equation only has the trivial solution, $U + \frac{3}{4}I$ is invertible.
    \item False. $A$ can have an eigenvalue $i$.
\end{enumerate}

Exercise 6.2.9

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Exercise 6.2.9

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