Rank equalities and inequalities for Kronecker products of matrices with applications. (Q1428183)

Let \(A_1,\dots ,A_k\) be square matrices of orders \(n_1,\dots ,n_k\), respectively, and denote \(A=A_1\otimes\cdots\otimes A_k\), \(n=n_1\dots n_k\), where \(A\otimes B\) is the Kronecker product of \(A=(a_{ij})_{m\times n}\) and \(B=(b_{ij})_{p\times q}\) over a field \({\mathbb F}\), i.e., \(A\otimes B=(a_{ij}B)\in{\mathbb F}^{mp\times nq}\). Several rank equalities and inequalities are established. The obtained results are applied to find upper and lower bounds for the dimension of the range of the linear transformations \(T_1(X)=X-AXB\) and \(T_2(X)=AX-XB.\) The following inequality is proved. Theorem: Let \(A=A_1\otimes\cdots\otimes A_k,\) and denote \(rk(I_{n_i}-A_i)=r_i\), \(i=1,\dots ,k\). Then \(rk(I_n-A)\leq n-(n_1-r_1)(n_2-r_2)\dots (n_k-r_k)\).