Let $X = \{|0\rangle,|1\rangle\}$ base of Hilbert space $H$ and $X' = \{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$ base $H^{\otimes2}$
I have quantum circuit with two hadamard gates working with register, which contain two qubites $|q0, q1\rangle = |q0\rangle \otimes |q1\rangle$.
I need to find matrix of operators in base $X'$ which is realizing this circuit. Is possible to multiply matrix of hadamard gate*matrix of hadamard gate and the result will be correct answer?
I also need to find eigenvalues and eigenvectors of $H^{\otimes2}$
Could someone help me?