Complex matrix into real matrix

Question

Can the matrix \begin{bmatrix}e^{-it}&e^{-it}\\ie^{it}&-ie^{-it}\end{bmatrix}

be turned into the matrix \begin{bmatrix}\cos t&\sin t\\-\sin t&\cos t\end{bmatrix}

I tried taking the euler definition of cosine and sine and breaking the first matrix into its constituents of cosine and sine, yet I can't seem to get the second column since they have with them imaginary numbers as coefficients, and multiplying by $i$ will cause the equivalence to shatter. Any suggestions?

Answer 1

It is not clear to me what is that you mean when you talk about turning a matrix into another matrix, but if you are talking about equivalent matrices, then the answer is negative. If the were equivalent, then they would have the same traces. But the trace of the original matrix is not, in general, a real number, whereas the trace of the second matrix is always a real number.