Is it possible to transform a metric tensor that has both 1's and −1's on the diagonal to a form in which only 1's are present?

Question

For example, can one change the coordinate system so that the metric tensor of signature $g_{\mu\nu} = diag(1,1,-1)$ is brought into the form $g_{\mu\nu} = diag(1,1,1)$? I guess not, because otherwise the Minkowsky metric could be reshaped to be completely positive, but I cannot find a good way to prove it.