### Orthogonal and symmetric maps: Orthogonal maps

### Orthogonal transformation matrices

We now discuss the role of matrices for general orthogonal maps.

Transition matrices and orthonormal bases If #\alpha# and #\beta# are two orthonormal bases in a real inner product space, then the transition matrix is #{}_\beta I_\alpha# is orthogonal and satisfies

\[

{}_\alpha I_\beta ={}_\beta I_\alpha^\top

\]

As a consequence, we have the following generalization of theorem *Orthogonality criteria for matrices* to arbitrary finite-dimensional inner product spaces.

Orthogonality criterion for linear mapsLet #\alpha# be an orthonormal basis for a real inner product space #V# of finite dimension and #L :V\rightarrow V# a linear map. Then #L# is orthogonal if and only if the matrix #L_\alpha# is orthogonal.

#S_{\varepsilon}= # #\dfrac{1}{9}\,\matrix{1 & -8 & -4 \\ -8 & 1 & -4 \\ -4 & -4 & 7 \\ }#

The system #\basis{\rv{2,2,1},\rv{-2,2,0},\rv{0,1,-2}}# is linearly independent and therefore a basis for #\mathbb{R}^3#.

From

\[\begin{array}{rcl} S_{\varepsilon} &=&{}_{\varepsilon} I_{\beta}\, S_\beta\,{}_\beta I_{\varepsilon}\\

&=&\matrix{2 & -2 & 0 \\ 2 & 2 & 1 \\ 1 & 0 & -2 \\ }\, \matrix{-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ }\,\matrix{2 & -2 & 0 \\ 2 & 2 & 1 \\ 1 & 0 & -2 \\ }^{-1}\\ &=&\matrix{-2 & -2 & 0 \\ -2 & 2 & 1 \\ -1 & 0 & -2 \\ }\,\matrix{{{2}\over{9}} & {{2}\over{9}} & {{1}\over{9}} \\ -{{5}\over{18}} & {{2}\over{9}} & {{1}\over{9}} \\ {{1}\over{9}} & {{1}\over{9}} & -{{4}\over{9}} \\ }\\ &=&\dfrac{1}{9}\,\matrix{1 & -8 & -4 \\ -8 & 1 & -4 \\ -4 & -4 & 7 \\ } \end{array}\]

The system #\basis{\rv{2,2,1},\rv{-2,2,0},\rv{0,1,-2}}# is linearly independent and therefore a basis for #\mathbb{R}^3#.

From

- #S(\rv{2,2,1}) = # #-\rv{2,2,1}#
- #S(\rv{-2,2,0}) = # #\rv{-2,2,0}#
- #S(\rv{0,1,-2}) = # #\rv{0,1,-2}#

\[\begin{array}{rcl} S_{\varepsilon} &=&{}_{\varepsilon} I_{\beta}\, S_\beta\,{}_\beta I_{\varepsilon}\\

&=&\matrix{2 & -2 & 0 \\ 2 & 2 & 1 \\ 1 & 0 & -2 \\ }\, \matrix{-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ }\,\matrix{2 & -2 & 0 \\ 2 & 2 & 1 \\ 1 & 0 & -2 \\ }^{-1}\\ &=&\matrix{-2 & -2 & 0 \\ -2 & 2 & 1 \\ -1 & 0 & -2 \\ }\,\matrix{{{2}\over{9}} & {{2}\over{9}} & {{1}\over{9}} \\ -{{5}\over{18}} & {{2}\over{9}} & {{1}\over{9}} \\ {{1}\over{9}} & {{1}\over{9}} & -{{4}\over{9}} \\ }\\ &=&\dfrac{1}{9}\,\matrix{1 & -8 & -4 \\ -8 & 1 & -4 \\ -4 & -4 & 7 \\ } \end{array}\]

Both #S_{\varepsilon}# and the (diagonal) matrix #S_\beta# with diagonal #-1,1,1# of #S# with respect to the basis \( \beta = \basis{\rv{2,2,1},\rv{-2,2,0},\rv{0,1,-2}}\) are orthogonal, but the transition matrix #{}_{\varepsilon} I_\beta# is not orthogonal. If we use the orthonormal basis \[\gamma=\basis{\left[ {{2}\over{3}} , {{2}\over{3}} , {{1}\over{3}} \right] ,\left[ -{{1}\over{\sqrt{2}}} , {{1}\over{\sqrt{2}}} , 0 \right] ,\left[ {{1}\over{3\cdot \sqrt{2}}} , {{1}\over{3\cdot \sqrt{2}}} , -{{2^{{{3}\over{2}}}}\over{3}} \right] }\] instead of #\beta#, then we find #S_\gamma = S_\beta#, but the transition matrix #{}_{\varepsilon} I_\gamma# is orthogonal.

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