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Some exercises on orthogonal matrices
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\problemno{1} Let $A$ be a square matrix with real entries.  Explain why
$A A^T = I$ \IFF\ $A$ has orthonormal columns.

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\problemno{2} Show that if $A$ is an orthogonal matrix and $\lambda$ is
a real eigenvalue of $A$, then $\lambda$ is $1$ or $-1$.

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\problemno{3} Show that if $A$ is a $2 \times 2$ rotation matrix, and 
$A \not= I$, then $A$ does not have a real eigenvalue.

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\problemno{4} Show that the eigenvalues of a square matrix $A$ are
equal to the eigenvalues of its transpose.  [{\sc Hint.}\ Consider
the characteristic equation $\det(\lambda I - A) = 0$.]

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\problemno{5} Show that if $\lambda$ is an eigenvalue of an orthogonal 
matrix, so is $\lambda^{-1}$.

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\problemno{6} Show that if $A$ is a $3 \times 3$ matrix, then $A$ has
a real eigenvalue.  If moreover $A$ is orthogonal and $\det(A) = 1$,
then $1$ is an eigenvalue of $A$.  Interpret this geometrically.