| Abstract: | Let Omega be a field, and let F denote the Frobenius matrix:$[F = left( {begin{array}{*{20}{c}}0&{ - {alpha _n}}{{E_{n - 1}}}&alpha end{array}} right)]$where alpha is an n-1 dimentional vector over Q, and E_n- 1 is identity matrix over Omega.Theorem 1. There hold two elementary decompositions of Frobenius matrix:(i) F=SJB,where S, J are two symmetric matrices, and B is an involutory matrix;(ii) F=CQD,where O is an involutory matrix, Q is an orthogonal matrix over Omega, and D is adiagonal matrix.We use the decomposition (i) to deduce the following two theorems:Theorem 2. Every square matrix over Omega is a product of twe symmetric matricesand one involutory matrix. Theorem 3. Every square matrix over Omega is a product of not more than foursymmetric matrices.By using the decomposition (ii), we easily verify the followingTheorem 4(Wonenburger-Djokovic') . The necessary and sufficient conditionthat a square matrix A may be decomposed as a product of two involutory matrices isthat A is nonsingular and similar to its inverse A^-1 over Q (See [2, 3]).We also use the decomosition (ii) to obtainTheorem 5. Every unimodular matrix is similar to the matrix CQB, whereC, B are two involutory matrices, and Q is an orthogonal matrix over Q.As a consequence of Theorem 5. we deduce immediately the followingTheorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may bedecomposed as a product of not more than four involutory matrices (See [1] ).Finally, we use the decomposition (ii) to derive the followingThoerem 7. If the unimodular matrix A possesses one invariant factor whichis not constant polynomial, or the determinant of the unimodular matrix A is I andA possesses two invariant factors with the same degree (>0), then A may bedecomposed as a product of three involutory matrices.All of the proofs of the above theorems are constructive. |
