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\begin{center}\tcr{\rmH Orthogonal Matrices} 
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\item[\tcr{Definition:}] An \mc{n\times n} matrix \mc{B = (\vec{b}_1|\vec{b}_2|\ldots|\vec{b}_n)} is called \tcr{orthogonal} if its column vectors \mc{\vec{b}_1, \vec{b}_2,\ldots,\vec{b}_n} form an \tcg{orthonormal} basis of \mc{\R^n}. 
 
\item[\tcr{Theorem 11:}] If \mc{B = (\vec{b}_1|\vec{b}_2|\ldots|\vec{b}_n)} is an \mc{n\times n} matrix, then the following conditions are equivalent: 

(1) \mc{B} is \tcg{orthogonal} (i.e.\ the column vectors \mc{\vec{b}_1,\vec{b}_2,\ldots,\vec{b}_n} are an \tcr{orthonormal basis} of \mc{\R^n};

(2) \mc{B^tB = I}, i.e.\ \mc{B^{-1} = B^t};

(3) \mc{(B\vec{v}\cdot B\vec{w}) = (\vec{v}\cdot\vec{w}),}  for all \mc{\vec{v},\vec{w}\in \R^n};

(4) \mc{||B\vec{v}|| = ||\vec{v}||}, for all \mc{\vec{v}\in \R^n}.
 
\item[\tcr{Note:}] Thus, the \tcg{orthogonal matrices} are precisely those that preserve \tcg{lengths} and \tcg{angles}
(when viewed as a \tcg{linear transformation}).

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