I was reading a research article on quantum computing and didn't understand the tensor notations for the unitary operations. The article defined two controlled gates.
Let $U_{2^m}$ be a $2^m \times 2^m$ unitary matrix, $I_{2^m}$ be a $2^m \times 2^m$ identity matrix. Then, controlled gates $C_n^j(U_{2^m})$ and $V_n^j(U_{2^m})$ with $n$ control qubits and $m$ target qubits are defined by $$ C_n^j(U_{2^m})=(|j\rangle \langle j|) \otimes U_{2^m}+ \sum_{i=0,i \neq j}^{2^n-1}((|i\rangle \langle i| \ \otimes I_{2^m}$$
$$ V_n^j(U_{2^m}) = U_{2^m} \otimes (|j\rangle \langle j|) + \sum_{i=0,i \neq j}^{2^n-1}( I_{2^m} \otimes (|i\rangle \langle i| ))$$ Then they say that $C_2^j(X)$ and $V_2^j(X) $are toffoli gates. Can someone explain the equations that are given and how does this special case be a Toffoli?