The question on the book is:
The Hadamard operator on one qubit may be written as $$\frac{1}{\sqrt{2}}[(|0\rangle + |1\rangle)\langle0| + (|0\rangle - |1\rangle)\langle1|]$$ Show explicitly that the Hadamard transform on $n$ qubits, $H^{\otimes n}$, may be written as $$H^{\otimes n}=\frac{1}{\sqrt{2^n}}\sum_{x,y}(-1)^{x\cdot y}|x\rangle\langle y|.\quad\quad \textrm{(2.55)}$$ Write out an explicit matrix representation for $H^{\otimes 2}$.
$H^{\otimes 2}$ is easy. It is $\frac{1}{2}\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 \end{bmatrix} $. It can be obtained as the Kronecker product of two $\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1\\ 1 & -1\\ \end{bmatrix}$. However, I am not sure how this is linked to (2.55). Say $|11\rangle\langle11|$. This term should has a coefficient of 1 according to the matrix of $H^{\otimes 2}$. (I think it is the bottom right corner.) However, according to (2.55), the coefficient seems to be -1 since $x=y=3$ and $x\cdot y=9$. In my understanding, the ranges for both $x$ and $y$ are $\{0, 1, 2, 3\}$.
Am I wrong with something? Do I understand the meanings of $x$, $y$, and $\cdot$ correctly? Or, is (2.55) wrong?