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, $(A^\mu)^1=L^\mu_{\nu} A^\nu$, (the exponent 1 on $A^\mu$ is just indicating that it's primed) where:

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

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

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

$\eta_{\mu\nu}= $$ \begin{matrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix}$

we could say that $(A_{\mu})^1= M_{\mu}^\nu A_{\nu}$. According to my calculations:

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

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

which doesn't agree with my calculations.

I'm sorry for my awfull notation and missing matrices' brackets (I'm still getting familiar with MathJax), I hope you can understand it and I'd be bery gratefull to anyone who can explain to me what I have done wrong.