Matrix completion theorems

Author:
Morris Newman

Journal:
Proc. Amer. Math. Soc. **94** (1985), 39-45

MSC:
Primary 15A33; Secondary 15A57

DOI:
https://doi.org/10.1090/S0002-9939-1985-0781052-8

MathSciNet review:
781052

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Abstract: Let be a principal ideal ring, the set of matrices over . The following results are proved:

(a) Let . Then the least nonnegative integer such that a matrix exists which belongs to is , where is the number of invariant factors of equal to 1.

(b) Any primitive element of may be completed to a symplectic matrix.

(c) If are such that is primitive and is symmetric, then may be completed to a symplectic matrix.

(d) If , are such that is primitive and is symmetric, then may be completed to a symmetric element of , provided that .

(e) If , then any primitive element of occurs as the first row of the commutator of two elements of .

**[1]**C. C. MacDuffee,*The theory of matrices*, Chelsea, New York, 1946.**[2]**M. Newman,*Integral matrices*, Academic Press, New York, 1972. MR**0340283 (49:5038)****[3]**-,*Symmetric completions and products of symmetric matrices*, Trans. Amer. Math. Soc.**186**(1973), 191-201. MR**0485931 (58:5725)****[4]**I. Reiner,*Symplectic modular complements*, Trans. Amer. Math. Soc.**77**(1954), 498-505. MR**0067076 (16:666e)****[5]**C. L. Siegel,*Über die analytische Theorie der quatratischen Formen*, Ann. of Math. (2)**36**(1935), 527-606. MR**1503238**

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0781052-8

Article copyright:
© Copyright 1985
American Mathematical Society