The Orthogonal Inner Product

A linear transformation  
  on a vector space  
  is orthogonal if for all vectors  
\[\mathbf{v} \in V\]
\[\left| T \mathbf{v} \right| = \left| \mathbf{v} \right|\]
All rotations and reflections are orthogonal, as are any sequence of rotations and reflections. Because the transformation  
  preserves the lengths or magnitudes of vectors, the associated matrix must have determinant 1 and the columns must have magnitude 1 and any two columns must be orthogonal and have dot product zero.
Example: Let a linear transformation have associated matrix  
\[\left( \begin{array}{cc} cos \theta & -sin \theta \\ sin \theta & cos \theta \end{array} \right)\]
This matrix represents a rotation of  
  anticlockwise by an angle  
The columns are the vectors  
\[\mathbf{v}_1 = \begin{pmatrix}cos \theta\\sin \theta \end{pmatrix} , \: \mathbf{v}_2 = \begin{pmatrix}- sin \theta\\cos \theta \end{pmatrix} \]
\[\left| \mathbf{v}_1 \right| = cos^2 \theta + sin^2 \theta =1\]

\[\left| \mathbf{v}_2 \right| = (-sin \theta)^2 + cos^2 \theta =1\]

\[ \mathbf{v}_1 \cdot \mathbf{v}_2 = cos \theta \times - sin \theta + sin \theta \times cos \theta =0\]

We can also define an inner product on  
\[\langle \mathbf{v}_1 , \mathbf{v}_2 \rangle = \mathbf{v}_1^T M \mathbf{v}_2\]
  is the matrix associated with the orthogonal transformation  
The inner product defined in this way has all the required properties, of being symmetric, positive definite and linear in both arguments.

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