Products of complic cosquares and pseudo-involutory matrices |
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Authors: | C S Ballntine |
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Institution: |
a Department of Mathematics, Oregon State University, Corvallis, OR |
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Abstract: | LetF be a field with (nontrivial) involution (i.e.F-conjugation). A nonsingular matrix Aover Fis called a complic F-cosquare provided A=S*-1for some matrix Sover Fand is called p.i. (pseudo-involutory) provided A=A-1 It is shown that Ais a complic F-cosquare iff Ais the product of two p.i. matrices over Fand that det (AA)=1 iff Ais the product of two complic F-cosquares (hence iff A is the product of four p.i. matrices over F). It is conjectured that, except for one obvious case (2 x 2 matrices over the field of order 2), every unimodular matrix A over an arbitrary field Fis a product S1ST:1T with S1 and Tover FThis conjecture is proved for matricesAof order ≤3. |
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