Symmetric completions and products of symmetric matrices
Author:
Morris Newman
Journal:
Trans. Amer. Math. Soc. 186 (1973), 191201
MSC:
Primary 15A33
MathSciNet review:
0485931
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Abstract: We show that any vector of n relatively prime coordinates from a principal ideal ring R may be completed to a symmetric matrix of , provided that . The result is also true for if R is the ring of integers Z. This implies for example that if F is a field, any matrix of is the product of a fixed number of symmetric matrices of except when , , which is a genuine exception.
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C. C. MacDuffee, The theory of matrices, Chelsea, New York, 1946.
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Morris
Newman, Integral matrices, Academic Press, New York, 1972.
Pure and Applied Mathematics, Vol. 45. MR 0340283
(49 #5038)
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Olga
Taussky, The role of symmetric matrices in the study of general
matrices, Linear Algebra and Appl. 5 (1972),
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Olga
Taussky, The factorization of an integral matrix into a product of
two integral symmetric matrices. I, Acta Arith. 24
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, The factorization of an integral matrix into a product of two integral symmetric matrices. II, Comm. Pure Appl. Math. (to appear).
 [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]
 O. Taussky, The role of symmetric matrices in the study of general matrices, Linear Algebra and Appl. 5 (1972), 147154. MR 0302674 (46:1818)
 [4]
 , The factorization of an integral matrix into a product of two integral symmetric matrices. I, Acta Arith. (to appear). MR 0335551 (49:332)
 [5]
 , The factorization of an integral matrix into a product of two integral symmetric matrices. II, Comm. Pure Appl. Math. (to appear).
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DOI:
http://dx.doi.org/10.1090/S00029947197304859317
PII:
S 00029947(1973)04859317
Keywords:
Principal ideal rings,
fields,
symmetric matrices,
unimodular matrices,
symmetric completion
Article copyright:
© Copyright 1973 American Mathematical Society
