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Permanents of direct products. (English) Zbl 0139.02602

It is well known that if \(A\) and \(B\) are \(n\) and \(m\)-square matrices, respectively, then \(\det(A\otimes B) = (\det A)^m (\det B)^n\), where \(A\otimes B)\) is the tensor or direct product of \(A\) and \(B\). This implies
\[ \vert\det(A\otimes B)\vert^2 = (\det(AA^*))^m (\det(B^*B))^n, \]
where \(A^*\) is the conjugate transpose of \(A\).
In the present paper the following results are proved:
\[ \vert\operatorname{per}(A\otimes B) \vert^2 \le (\operatorname{per}(AA^*))^m (\operatorname{per}(B^*B))^n. \tag{I} \]
Equality holds if and only if \(A\) has a zero row or \(B\) has a zero column, or \(A\) and \(B\) are both generalized permutation matrices, that is, each is a product of a diagonal matrix and a permutation matrix.
(II) If \(A\) and \(B\) are positive semidefinite hermitian then
\[ \operatorname{per}(A\otimes B) \ge (1/n!)^m (1/m!)^n (\operatorname{per} A)^m (\operatorname{per}B)^m. \]
Finally a combinatorial application of (I) is obtained.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
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References:

[1] R. A. Brualdi, Permanent of the direct product of matrices, Abstract 64T-603, Notices Amer. Math. Soc. 11 (1964), 770.
[2] C. C. MacDuffee, The theory of matrices, Chelsea, New York, 1946. · Zbl 0007.19507
[3] Marvin Marcus and Morris Newman, Inequalities for the permanent function, Ann. of Math. (2) 75 (1962), 47 – 62. · Zbl 0103.00703 · doi:10.2307/1970418
[4] Marvin Marcus, The Hadamard theorem for permanents, Proc. Amer. Math. Soc. 15 (1964), 967 – 973. · Zbl 0166.29903
[5] Herbert John Ryser, Combinatorial mathematics, The Carus Mathematical Monographs, No. 14, Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York, 1963. · Zbl 0302.05001
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