A decomposition depended on directions for nonsingular linear transformation

In this paper, we give a decomposition depending on p(1pn–2) orthonormal directions assigned for nonsingular linear transformation F on a n-dimension (n3) Euclidean space E_n, and then prove that there exist q(q=n–p) quasi-principal directions for F depending on the preceding p orthonormal directions. As applicance of the preceding result, we derive that there exist at least two orthonormal principal directions of strain in arbitrary plane of body which is in homogeneous deformation, and strain energy density is function of 5 real numbers under arbitrary quasi-principal base for the preceding nonsingular linear transformation.