# orthogonal matrices

A real square $n\times n$ matrix $Q$ is orthogonal^{} if ${Q}^{\mathrm{T}}Q=I$, i.e., if ${Q}^{-1}={Q}^{\mathrm{T}}$. 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: ${(Qx)}^{\mathrm{T}}Qy={x}^{\mathrm{T}}{Q}^{\mathrm{T}}Qy={x}^{\mathrm{T}}y$, 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 ${Q}^{\mathrm{T}}Ax\approx {Q}^{\mathrm{T}}b$.

Orthogonal matrices can be thought of as the real case of unitary matrices^{}. A unitary matrix $U\in {\u2102}^{n\times n}$ has the property ${U}^{*}U=I$, where ${U}^{*}=\overline{{U}^{\mathrm{T}}}$ (the conjugate transpose^{}). Since $\overline{{Q}^{\mathrm{T}}}={Q}^{\mathrm{T}}$ 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:

$$\left(\begin{array}{cc}\hfill \mathrm{cos}(\alpha )\hfill & \hfill \mathrm{sin}(\alpha )\hfill \\ \hfill -\mathrm{sin}(\alpha )\hfill & \hfill \mathrm{cos}(\alpha )\hfill \end{array}\right)\text{or}\left(\begin{array}{cc}\hfill \mathrm{cos}(\alpha )\hfill & \hfill \mathrm{sin}(\alpha )\hfill \\ \hfill \mathrm{sin}(\alpha )\hfill & \hfill -\mathrm{cos}(\alpha )\hfill \end{array}\right)$$ |

respectively.

This entry is based on content from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)

## References

- 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.

Title | orthogonal matrices |
---|---|

Canonical name | OrthogonalMatrices |

Date of creation | 2013-03-22 12:05:19 |

Last modified on | 2013-03-22 12:05:19 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 11 |

Author | akrowne (2) |

Entry type | Definition |

Classification | msc 15-00 |

Related topic | OrthogonalPolynomials |

Related topic | RotationMatrix |