Suppose we have a vector $A^\mu$. This vector has 4 components (0 through 3). If we consider the Lorentz's transformation of each component, we could simplify them into matrix notation, ${A^\prime}^\mu=L^\mu_{\nu} A^\nu$, where:

$$L_{\nu}^\mu= \left( \begin {matrix}\gamma & -v\gamma & 0 & 0\\ -v\gamma & \gamma & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix}\right)$$

where $\gamma=(1-\frac{v^2}{c^2})^{-\frac{1}{2}}$

Knowing that $A_{\mu}= \eta_{\mu\nu} A^\nu$ and

$$\eta_{\mu\nu}= \left( \begin{matrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix}\right)$$

we could say that $A_\mu^\prime= M_{\mu}^\nu A_{\nu}$. According to my calculations:

$$M_{\mu}^\nu= \left( \begin{matrix} -\gamma&v\gamma&0&0\\ v\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix}\right)$$

But according to Leonard Susskind's lectures: $M=\eta L\eta$

which doesn't agree with my calculations.