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- W2032020327 abstract "ln a recent number of the Transactions of the Connecticut Acadelny, E. B. MRILSON obtains a necessary and sufficient condition that a linear transformation be factorable into two involutory transformations. If this condition is translated from the language of dyadics, which WILSON uses, into that of the algebra of matrices, its form suggests at once a theorem of FROBENIUS con. cerning transformations of a bilinear form into itself. A mere combination of these two theorems is sufficient to establish the following, which is simpler in statement than either: I. A necessary and sufScient condition that a lineat transforr)tatton be such as to carry some non-singulclr bilinear fornx into itself, iys that it be factorstble bnto two involutory transformations. It will be seen that the part of this theorem relating to the necessity of the conclition is a generalization of a theorem of P. F. SMITH t which VVrILsoN uses in proving his theorem, and which is substantially as follows: II. (SMITH). A necessary cond:tionthat a linear transformation be such as to carry some non-sing7xlar q?badfratic fortn into itself, iys that it be fcletorable into two involutory transformations. The converse of SMITH'S theorem is not true, as is remarked by WILSON; a substitution which is factorable into two involutory transformations is not necessarily capable of carrying a non-singular qt6adratic form into itself. In fact, the question of the possibility of factoring a transformation into two involutory transformations has no essential relatioll to the subject of symmetric bilinear forms, as distinguished from bilinear forlns in general. :§:" @default.
- W2032020327 created "2016-06-24" @default.
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- W2032020327 date "1909-04-01" @default.
- W2032020327 modified "2023-09-26" @default.
- W2032020327 title "Resolution into involutory substitutions of the transformations of a non-singular bilinear form into itself" @default.
- W2032020327 doi "https://doi.org/10.1090/s0002-9947-1909-1500850-3" @default.
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