By Zeldin S. D.

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The most general transformation V which connects the involutional matrices A and B of the lemma via a similarity transformation V À1 AV ˆ B is given by V ˆ FA Y ˆ YFB (2X1X6) where FA and FB are the same function of A and B, respectively. 1 Involutional transformations 19 Remark. If both A and B are IUH matrices, then so is Y ; however, V is unitary but not involutional in general, for V 2 ˆ AB. In two dimensions, the involutional matrices A, B and Y are all improper whereas V is proper, being a product of two improper matrices.

1. The most general transformation V which connects the involutional matrices A and B of the lemma via a similarity transformation V À1 AV ˆ B is given by V ˆ FA Y ˆ YFB (2X1X6) where FA and FB are the same function of A and B, respectively. 1 Involutional transformations 19 Remark. If both A and B are IUH matrices, then so is Y ; however, V is unitary but not involutional in general, for V 2 ˆ AB. In two dimensions, the involutional matrices A, B and Y are all improper whereas V is proper, being a product of two improper matrices.

E. AB ‡ BA ˆ 2c1, c Tˆ À1 then there exists an involutional transformation that interchanges A and B via YAY ˆ B or A ˆ YBY ; Y2 ˆ 1 (2X1X4) where Y ˆ (A ‡ B)a(2 ‡ 2c)1a2 (2X1X5) The proof is elementary. 4). 1. The most general transformation V which connects the involutional matrices A and B of the lemma via a similarity transformation V À1 AV ˆ B is given by V ˆ FA Y ˆ YFB (2X1X6) where FA and FB are the same function of A and B, respectively. 1 Involutional transformations 19 Remark. If both A and B are IUH matrices, then so is Y ; however, V is unitary but not involutional in general, for V 2 ˆ AB.

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