Exercise 4.2.1

Question

$
ewcommand{\T}{^\mathsf{T}}$
Let $A$ be $n 	imes n$ matrix. True or false:
\begin{enumerate}[label = \alph*)]
    \item $A\T$ has the same eigenvalues as $A$.
    \item $A\T$ has the same eigenvectors as $A$.
    \item If A is diagonalizable, then so is $A\T$.
\end{enumerate}

Jing Huang · Accepted Answer

\begin{enumerate}[label = \alph*)]
    \item True, $\det(A - \lambda I) = \det (A - \lambda I)\T = \det (A\T - \lambda I)$
    \item False.
    \item True, $A = SDS^{-1}, A\T = (SDS^{-1})\T = (S^{-1})\T D\T S\T = (S\T)^{-1} D  S\T.$ 
\end{enumerate}

Exercise 4.2.1

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Exercise 4.2.1

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