XML:
objects1176orthogonal matrices is owned by Aaron Krowne. Aaron Krowne. "orthogonal matrices" (version 7). PlanetMath.org. Freely available at http://planetmath.org/OrthogonalMatrices.html. 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. This entry is based on content from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html) Bibliography 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997. NS:published, NS:Section:Reference AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works (handbooks, dictionaries, bibliographies, etc.)) Contributors | History | Req. Co-author | Watch | Style: Flat Threaded Expand: all none 1 2 3 4 5 6 7 8 9 Order: Oldest First Newest first Q^T Q = Id <=> Q^-1 = Q^T ? by matte on 2005-04-10 07:27:59 Hi An inverse of A is matrix A^-1 such that A A^-1 = A^-1 A = Id. In this entry, does it follow from Q^T Q = Id that Q^T = Q^-1? Or more generally, is it possible that A B = Id for some matrices A, B, but B A \neq Id? Matte [ reply | up ] Re: Q^T Q = Id <=> Q^-1 = Q^T ? by Serg on 2005-04-10 08:31:49 Re: Q^T Q = Id <=> Q^-1 = Q^T ? by matte on 2005-04-10 09:22:59 View style: jsMath HTML HTML with images PDF file page images TeX source None. [ View all 5 ] Post Comment | Attach New Article | Suggest Correction
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.
This entry is based on content from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)
