Additive Decomposition of Real Matrices

The orthogonal orbit ${cal O}(A)$ of an n × n real matrix A is the set of real matrices of the form $P^t AP$ where $P^t P = I_n$ . We show that $A/ | A|$ is an affine sum of four orthogonal matrices, and note that $A^t$ can always be written as an affine combination of no more than 2 n m 1 matrices in ${cal O}(A)$ . This improves some recent results of Zhan, and answers some of his questions. Other related results are also discussed.