In a homework problem, we are asked to find the eigenvalues of a linear operator $T(X)= A(X)^t A$ ($A$ is a matrix, $X^t$ is $X$'s transpose). My calculation of $T(x)$ lead me to a seemingly answer, but on second thought, why is this problem possible? An eigenvector is supposed to stay on its own direction, but my calculation just shows how the "transpose of the vector" is scaled, which doesn't make any sense.