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orthogonal matrices

(Definition)

A real square $n \times n$ matrix 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 $\vert\vert Qx\vert\vert _2 = \vert\vert x\vert\vert _2$ . 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 $U \in \mathbb{C}^{n \times n}$ has the property , where $U^* = \overline{U^{\mathrm T}}$ (the conjugate transpose). Since $\overline{Q^{\mathrm T}} = Q^{\mathrm T}$ for real , orthogonal matrices are unitary.

An orthogonal matrix 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:

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

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.

"orthogonal matrices" is owned by akrowne.

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See Also: orthogonal polynomials, rotation matrix

Attachments:: decomposition of orthogonal operators as rotations and reflections (Theorem) by stevecheng
derivation of 2D reflection matrix (Derivation) by stevecheng
characteristic polynomial of a orthogonal matrix is a reciprocal polynomial (Theorem) by matte

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Cross-references: reflections, rotations, rounding errors, Householder transformations, Givens rotations, unitary, conjugate transpose, property, unitary matrices, equivalent, least squares problem, Euclidean norm, Transform, inner products, linear algebra, orthonormal basis, columns, rows, orthogonal, matrix, real
There are 23 references to this entry.

This is version 7 of orthogonal matrices, born on 2002-01-02, modified 2006-11-11.
Object id is 1176, canonical name is OrthogonalMatrices.
Accessed 26110 times total.

Classification:

AMS MSC:

15-00 (Linear and multilinear algebra; matrix theory :: General reference works )

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