Find the linear maps that preserve this bilinear form. This time I want to know which linear maps are the ones that preserve the following bilinear form:
$$\beta(x,y)=2(x_1y_1-x_2y_2)$$
and they give as a hint the matrix:
$$\begin{pmatrix}
cosh&sinh\\-sinh&cosh
\end{pmatrix}$$
then when I multiply this matrix by the one generated by the bilinear form, this is: 
$$\begin{pmatrix}
2&0\\0&-2
\end{pmatrix}\begin{pmatrix}
cosh&sinh\\-sinh&cosh
\end{pmatrix}$$
Then I get a matrix that has the same shape as the one of the bilinear form, but I think this is not preservation, so Can someone help me with this problem please? (I have tried to compute the determinants but I think this has nothing to do with the bilinear form)
Thanks a ot in advance :). 
 A: Under the linear transformation $x\to Ax$ we have
$$x = \left(\matrix{x_1\\x_2}\right) \to \left(\matrix{A_{11}x_1 + A_{12}x_2\\A_{21}x_1 + A_{22}x_2}\right)$$
so the bi-linear form transforms as
$$x_1y_1-x_2y_2 \to (A_{11}x_1 + A_{12}x_2)(A_{11}y_1 + A_{12}y_2) - (A_{21}x_1 + A_{22}x_2)(A_{21}y_1 + A_{22}y_2)$$
From this we see that the form is preserved iff
$$\matrix{A_{11}^2-A_{21}^2 &= 1\\A_{22}^2-A_{12}^2 &= 1\\A_{11}A_{12}-A_{21}A_{22} &= 0}$$
Since $A_{11}^2,A_{22}^2\geq 1$ we can take $A_{11}=\epsilon_{11}\cosh\alpha$ and $A_{22} = \epsilon_{22}\cosh\beta$ for some $\alpha,\beta\in\mathbb{R}$ and where $\epsilon_{ij}=\pm 1$. The identity $\cosh^2(x) - \sinh^2(x) = 1$ applied to the first two equations determines $A_{12}$ and $A_{21}$ (up to a sign) and the final equation fixes $\alpha=\pm\beta$ and gives a relation between the allowed combinations of signs $\epsilon_{ij}$. 

Any solution $A$ can be written on one of the four following forms:
$$\left\{\pm\left(\matrix{+c & +s\\ +s & +c}\right),~~~\pm\left(\matrix{+c & +s\\ -s & -c}\right)\right\}$$
where $s=\sinh(\alpha)$ and $c=\cosh(\alpha)$ for some $\alpha\in\mathbb{R}$.
