1.4.28 Matrix transformations

condition: Prove that the formula \( \quad \varphi(X)=A^{T} X A \) defines a linear transformation of the space of symmetric matrices \( \left(X^{T}=X\right) \). Find the Jordan basis and the Jordan form of the transformation. \[ A=\left(\begin{array}{cc} 3 & 4 \\ -1 & -1 \end{array}\right) \]