To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Orthogonal transformation

From Wikipedia, the free encyclopedia

In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we have[1]

Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases.

Orthogonal transformations are injective: if then , hence , so the kernel of is trivial.

Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The matrices corresponding to proper rotations (without reflection) have a determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions.

In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V.

If an orthogonal transformation is invertible (which is always the case when V is finite-dimensional) then its inverse is another orthogonal transformation. Its matrix representation is the transpose of the matrix representation of the original transformation.

YouTube Encyclopedic

  • 1/3
    64 837
    479 711
    47 030
  • Orthogonal matrices preserve angles and lengths | Linear Algebra | Khan Academy
  • Linear transformations and matrices | Essence of linear algebra, chapter 3
  • Example using orthogonal change-of-basis matrix to find transformation matrix | Khan Academy



Consider the inner-product space with the standard euclidean inner product and standard basis. Then, the matrix transformation

is orthogonal. To see this, consider


The previous example can be extended to construct all orthogonal transformations. For example, the following matrices define orthogonal transformations on :

See also


  1. ^ Rowland, Todd. "Orthogonal Transformation". MathWorld. Retrieved 4 May 2012.
This page was last edited on 12 July 2021, at 01:22
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.