Let A be an n X n matrix over a field of characteristic 2. If n is odd, then A is similar to an s-symmetric matrix (one symmetric around the diagonal from lower left to upper right). If n is even, this holds iff the elementary divisors of A that are odd powers of separable polynomials occur with even multiplicity. The proof uses the structure theory for pairs consisting of an inner product and a self-adjoint mapping for that inner product.
An n x n matrix A over a field k is called s-symmetric if it is symmetric around the secondary diagonal (upper right to lower left), i.e. Aij = A n+l-j,n+l-i'
Reversing the order of the rows, or the order of the columns, will give a symmetric matrix; but (contrary to what is said e.g. in Muir's book [4, p. 181) reversing both will preserve s-symmetry, and thus s-symmetric matrices are not necessarily similar to symmetric matrices. Indeed, when k is any field with char(k) # 2, A. Lee [ 1,2] has recently proved that s-symmetric matrices can have arbitrarily prescribed elementary divisors. In other words, any matrix over k is similar to an s-symmetric matrix. This paper completes the topic by settling the corresponding question in characteristic 2. We shall see that the theorem remains true when n is odd, while for even n there are certain parity restrictions on the elementary divisors.
As in Lee's work, we begin by noting that the s-symmetric matrices are precisely those that are selfadjoint with respect to the nondegenerate sym-