$A_{v}^{-1}$ = $A_{v}^{T}$
$A_{v}$ = Allignment matrix that alligns vector v with z axis
$=$\begin{bmatrix}
\frac{\lambda}{|v|} & \frac{-ab}{\lambda|v|} & \frac{-ac}{\lambda|v|} & 0 \\
0 & \frac{c}{|v|} & \frac{-b}{|v|} & 0 \\
\frac{a}{|v|} & \frac{b}{|v|} & \frac{c}{|v|} & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
here, $\lambda = \sqrt{b^2+c^2} $
$|v| = \sqrt{a^2+b^2+c^2} $
And vector $v = (a, b, c)$
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$\begingroup$ Are you asking "Why is the inverse of a rotation matrix the same as its transpose"? $\endgroup$– Simon FCommented Jan 5, 2018 at 17:08
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1$\begingroup$ That is a rotation or reflection matrix. You can check that the four columns are orthogonal to each other (the dot product of the columns is zero). Also they are of unit length, so the determinant is $\pm$1. A well known property of orthonormal square matrices is that transpose is equal to the inverse. $\endgroup$– Mauricio Cele Lopez BelonCommented Jan 5, 2018 at 21:26
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$\begingroup$ @MauricioCeleLopezBelon your explanation was truly helpful. Thanks. $\endgroup$– BhattacharjeeCommented Jan 6, 2018 at 6:51
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1 Answer
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To align vectors, you are applying a rotation.
If you consider a simple rotation matrix, e.g. rotation of angle $\theta$ around Z axis, then you will trivially see that its inverse matrix, a rotation of $-\theta$, is the transpose, since $sin(-\theta)=-sin(\theta)$ and $cos(-\theta)=cos(\theta)$.
An arbitrary rotation matrix, R, can be constructed by multiplication of other rotations, e.g $R = A \cdot B$. If the inverse of A and B exist, then $R^{-1} = B^{-1} \cdot A^{-1}$.
Similarly we know $(A \cdot B)^T = (B^T \cdot A^T)$.
Put these together and you have your answer.
