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Rank equalities and inequalities for Kronecker products of matrices with applications. (English) Zbl 1041.15016

Let A 1 ,,A k be square matrices of orders n 1 ,,n k , respectively, and denote A=A 1 A k , n=n 1 n k , where AB is the Kronecker product of A=(a ij ) m×n and B=(b ij ) p×q over a field 𝔽, i.e., AB=(a ij B)𝔽 mp×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 A k , and denote rk(I n i -A i )=r i , i=1,,k. Then rk(I n -A)n-(n 1 -r 1 )(n 2 -r 2 )(n k -r k ).

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