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Rank equalities and inequalities for Kronecker products of matrices with applications. (English) Zbl 1041.15016
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 ${\Bbb F}$, i.e., $A\otimes B=(a_{ij}B)\in{\Bbb F}^{mp\times nq}$. 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 $rk(I_n-A)\leq n-(n_1-r_1)(n_2-r_2)\dots (n_k-r_k)$.

MSC:
15A45Miscellaneous inequalities involving matrices
15A03Vector spaces, linear dependence, rank
15A04Linear transformations, semilinear transformations (linear algebra)
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References:
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