Orthogonal Transformation

An Orthogonal Transformation is a matrix tranformation that preserves length of vectors and angles between them.

• Example(s):
  • a mapping [math]\displaystyle{ T:A \to B }[/math] can be a matrix which acts as orthogonal transformation.

    Let there is an input matrix [math]\displaystyle{ A=[a_1, a_2]=\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix} }[/math].Here orthogonal transformation, that is a matrix [math]\displaystyle{ T=\begin{bmatrix}0 & -1 \\1 & 0 \end{bmatrix} }[/math] can be used to transform all input vectors to their respective orthogonal vectors without making any change in the length of vectors and angles between them. Here a matrix [math]\displaystyle{ B=[b_1, b_2] }[/math] is desired as output such that [math]\displaystyle{ a_1*b_1=0 }[/math] and [math]\displaystyle{ a_2*b_2=0 }[/math]

    [math]\displaystyle{ a_1=\begin{bmatrix}1 \\1 \end{bmatrix}; |a_1|=\sqrt{1^2+1^2}=\sqrt{2};\theta_1=arg(a_1)=\tan^{-1}(\frac{1}{1})=45^o }[/math]

    [math]\displaystyle{ a_2=\begin{bmatrix}1 \\0 \end{bmatrix}; |a_2|=\sqrt{1^2+0^2}=1;\theta_2=arg(a_2)=\tan^{-1}(\frac{0}{1})=0^o }[/math]

    The angle between [math]\displaystyle{ a_1 }[/math] and [math]\displaystyle{ a_2 }[/math] is [math]\displaystyle{ 45^o }[/math].

    Now applying orthogonal transformation [math]\displaystyle{ T }[/math] to [math]\displaystyle{ a_1 }[/math] and [math]\displaystyle{ a_2 }[/math] the respective orthogonal vectors [math]\displaystyle{ b_1 }[/math] and [math]\displaystyle{ b_2 }[/math] can be obtained.

    [math]\displaystyle{ Ta_1=b_1: \begin{bmatrix}0 & -1 \\1 & 0 \end{bmatrix}\begin{bmatrix}1 \\1 \end{bmatrix}=\begin{bmatrix}-1 \\1 \end{bmatrix} }[/math]

    [math]\displaystyle{ Ta_2=b_2: \begin{bmatrix}0 & -1 \\1 & 0 \end{bmatrix}\begin{bmatrix}1 \\0 \end{bmatrix}=\begin{bmatrix}0 \\1 \end{bmatrix} }[/math]

    [math]\displaystyle{ b_1=\begin{bmatrix}-1 \\1 \end{bmatrix}; |b_1|=\sqrt{(-1)^2+1^2}=\sqrt{2};\theta_3=arg(b_1)=\tan^{-1}(\frac{1}{-1})=135^o }[/math]

    [math]\displaystyle{ b_2=\begin{bmatrix}0 \\1 \end{bmatrix}; |b_2|=\sqrt{0^2+1^2}=1;\theta_4=arg(b_2)=\tan^{-1}(\frac{1}{0})=90^o }[/math]

    

    .

    Here it can be observed that the [math]\displaystyle{ a_1 }[/math] and its transformed orthogonal matrix [math]\displaystyle{ b_1 }[/math] both have the same lengths. Also [math]\displaystyle{ a_2 }[/math] and its transformed orthogonal matrix [math]\displaystyle{ b_2 }[/math] both have the same lengths. The angle between [math]\displaystyle{ a_1 }[/math] and [math]\displaystyle{ a_2 }[/math] is same as the angle between [math]\displaystyle{ b_1 }[/math] and [math]\displaystyle{ b_2 }[/math].

• See: Orthogonal Matrix, Linear Transformation, Nondegenerate Form, Symmetric Bilinear Form, Orthonormal Basis, Euclidean Space, Rotation (Mathematics), Reflection (Mathematics), Improper Rotation, Determinant.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/orthogonal_transformation Retrieved:2015-2-16.
  • In linear algebra, an 'orthogonal transformation is a linear transformation on a vector space V that has a nondegenerate symmetric bilinear form such that T preserves the bilinear form. That is, for each pair of elements of V, we have  :[math]\displaystyle{ \langle u,v \rangle = \langle Tu,Tv \rangle \, . }[/math]

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

    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 exchange left and right, similar to mirror images. The matrices corresponding to proper rotations (without reflection) have determinant +1. Transformations with reflection are represented by matrices with determinant −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.

    The inverse of an orthogonal transformation is another orthogonal transformation. Its matrix representation is the transpose of the matrix representation of the original transformation.

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