| 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 a
diagonal 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 matrices
and one involutory matrix.
Theorem 3. Every square matrix over \Omega is a product of not more than four
symmetric matrices.
By using the decomposition (ii), we easily verify the following
Theorem 4(Wonenburger-Djokovic') . The necessary and sufficient condition
that a square matrix A may be decomposed as a product of two involutory matrices is
that A is nonsingular and similar to its inverse A^-1 over Q (See 2, 3]).
We also use the decomosition (ii) to obtain
Theorem 5. Every unimodular matrix is similar to the matrix CQB, where
C, B are two involutory matrices, and Q is an orthogonal matrix over Q.
As a consequence of Theorem 5. we deduce immediately the following
Theorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may be
decomposed as a product of not more than four involutory matrices (See 1] ).
Finally, we use the decomposition (ii) to derive the following
Thoerem 7. If the unimodular matrix A possesses one invariant factor which
is not constant polynomial, or the determinant of the unimodular matrix A is I and
A possesses two invariant factors with the same degree (>0), then A may be
decomposed as a product of three involutory matrices.
All of the proofs of the above theorems are constructive. |
