Block determinants, partial determinants and the exponential map (Q6649805)

Let \(E_{ij}\) (\(i,j=1,\dots,m\)) be the standard basis for the matrix algebra \( M_{m}(\mathbb{C})\) and let \(F_{kl}\) (\(k,l=1,\dots,n\)) be the corresponding basis for \(M_{n}\mathbb{C})\). The Kronecker product of \(S=[s_{ij}]\in M_{m}( \mathbb{C})\) and \(T=[t_{kl}]\in M_{n}(\mathbb{C})\) is given as a block matrix by \(S\otimes T:=[s_{ij}T]_{i,j=1}^{m}\). Every matrix \(A\in \) \(M_{mn}( \mathbb{C})\) can be written in two forms:\N\[\NA=\sum_{k,\ell }A_{k\ell }\otimes F_{k\ell }=\sum_{i,j}E_{ij}\otimes A_{ij}^{^{\prime }},\N\]\Nwhere \(A_{ij}^{^{\prime }}\in M_{n}(\mathbb{C})\) and \(A_{k\ell }\in M_{m}( \mathbb{C}).\)\N\NThe tensor algebra \(M_{m}(\mathbb{C)\otimes }M_{n}(\mathbb{C})\) consists of all sums of products of the form \(S\otimes T\ \)and is isomorphic with both \( M_{mn}(\mathbb{C})\) and \(M_{n}(\mathbb{C)\otimes }M_{m}(\mathbb{C})\) in natural ways. Using matrix exponentials we also have \(e^{B}\otimes e^{C}=e^{B\oplus C}\) for \(B\in M_{m}\) and \(C\in M_{n}\) where \(B\oplus C:=B\otimes I+I\otimes C\). The partial determinants of \(A\) are defined by \( \det_{1}(A):=\sum \det (A_{k\ell })F_{k\ell }\) and \(\det_{2}(A):=\sum \det (A_{i,j}^{\prime })E_{ij}\) and partial traces are analogous. Both have been studied, but whereas the trace of a partial trace returns the trace of the original matrix, the analogous result is not generally true for determinants. In the present paper the authors modify the definition of determinant to obtain a function \(\mathrm{Det}\) which behaves better with respect to the tensor decomposition.