Say I have a Direction Cosine Matrix $[a]$ given by: $$ [a]= a_{ij}=cos(x_{i},x_{j}^{'})=cos(\vec{x},\vec{x}')= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} $$ In other words, the entries $a_{ij}$ are the cosines of the angles between the axes of the old coordinate system, $x_i$, and the new coordinate system, $x_{j}^{'}$, after being transformed by $[a]$. In other words:
$$\vec{x}'=x_{j}^{'}=a_{ij}x_{i}=[a]\vec{x}$$
If the axes of $\vec{x}'$ are orthogonal, then I believe $[a]$ is orthogonal. Regardless, assuming that $[a]$ is orthogonal, the entries of $[a]$ are not independent. Since $[a]^{-1}=[a]^{T}$ for an orthogonal matrix and $[a]^{-1}[a]=[a][a]^{-1}=[I]$ for invertible $[a]$, then $[a]^{T}[a]=[a][a]^{T}=[I]$. This leads to 12 distinct equations: $$ a_{11}^2+a_{21}^2+a_{31}^2=1 $$ $$ a_{12}^2+a_{22}^2+a_{32}^2=1 $$ $$ a_{13}^2+a_{23}^2+a_{33}^2=1 $$ $$ a_{11}^2+a_{12}^2+a_{13}^2=1 $$ $$ a_{21}^2+a_{22}^2+a_{23}^2=1 $$ $$ a_{31}^2+a_{32}^2+a_{33}^2=1 $$
$$ a_{11}a_{12}+a_{21}a_{22}+a_{31}a_{32}=0 $$ $$ a_{11}a_{13}+a_{21}a_{23}+a_{31}a_{33}=0 $$ $$ a_{12}a_{13}+a_{22}a_{23}+a_{32}a_{33}=0 $$ $$ a_{11}a_{21}+a_{12}a_{22}+a_{13}a_{23}=0 $$ $$ a_{11}a_{31}+a_{12}a_{32}+a_{13}a_{33}=0 $$ $$ a_{21}a_{31}+a_{22}a_{32}+a_{23}a_{33}=0 $$ Is there a way to take these relations into account in the original matrix $[a]$? I am unsure how to go about this since it is a nonlinear system of equations. Is there a general way to go about such a problem?