15-00-OrthogonalMatrices.tex

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A real square $n \times n$ matrix $Q$ is orthogonal if $Q^\trT Q = I$, i.e., if $Q^{-1} = Q^\trT$.  The rows and columns of an orthogonal matrix form an orthonormal basis.

Orthogonal matrices play a very important role in linear algebra.  Inner products are preserved under an orthogonal transform: $(Qx)^\trT Qy=x^\trT Q^\trT Qy=x^\trT y$, and also the Euclidean norm $||Qx||_2 = ||x||_2$.  An example of where this is useful is solving the least squares problem $Ax \approx b$ by solving the equivalent problem $Q^\trT Ax \approx Q^\trT b$.

Orthogonal matrices can be thought of as the real case of unitary matrices.  A unitary matrix $U \in \mathbb{C}^{n \times n}$ has the property $U^*U = I$, where $U^* = \overline{U^\trT}$ (the conjugate transpose).  Since $\overline{Q^\trT} = Q^\trT$ for real $Q$, orthogonal matrices are unitary.

An orthogonal matrix $Q$ has $\det(Q) = \pm 1$.

Important orthogonal matrices are Givens rotations and Householder transformations.  They help us maintain numerical stability because they do not amplify rounding errors.

Orthogonal $2 \times 2$ matrices are rotations or reflections if they have the form:

$$ \begin{pmatrix} \cos(\alpha) & \sin(\alpha) \\ -\sin(\alpha) & \cos(\alpha) \end{pmatrix}  \text{or} \begin{pmatrix} \cos(\alpha) & \sin(\alpha) \\ \sin(\alpha) & -\cos(\alpha) \end{pmatrix} $$

respectively.

{\it This entry is based on content from The Data Analysis Briefbook (\PMlinkexternal{http://rkb.home.cern.ch/rkb/titleA.html}{http://rkb.home.cern.ch/rkb/titleA.html})}
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\bibitem{Friedberg} Friedberg, Insell, Spence. {\it Linear Algebra}. Prentice-Hall Inc., 1997.

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