Why there is a factor 2/3 added to Clarke Transform matrix?

Question

Apparently, the matrix to transform the 3 vectors \$ U_a , U_b, U_c \$ into \$ U_\alpha, U_\beta \$ is: $$ \begin{bmatrix} U_{\alpha} \\ U_{\beta} \\ U_{0} \\ \end{bmatrix} = \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \end{bmatrix} \begin{bmatrix} U_{a} \\ U_{b} \\ U_{c} \\ \end{bmatrix} $$

Why in Clarke Transform, matrix is multiply by \$\frac{2}{3} \$

$$ T_{\alpha \beta 0} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \end{bmatrix} $$

Answer 1

If you do the transform without the 2/3 scale factor, the amplitude of the alpha-beta variables is 1.5 times higher than that of the ABC variables. The scaling is done only to maintain the amplitude across the transform. For example, taking a balanced 3-phase system having amplitude 1, the first row becomes $$ \cos{\omega t}-\frac{1}{2}\cos{(\omega t+\frac{2\pi}{3})}-\frac{1}{2}\cos{(\omega t-\frac{2\pi}{3})} $$ Using the identity $$ \cos{A}\cos{B} = \frac{1}{2}\cos{(A+B)} + \frac{1}{2}\cos{(A-B)} $$ this is reduced to $$ \cos{\omega t} - \cos{(\frac{2\pi}{3}})\cos{\omega t} = \frac{3}{2}\cos{\omega t} $$