Abstract: Let A_i be m_i \times n matrix i=1, \cdots, t, and B_1, \cdots B_I be symmetric n \times n matrices. For the case of m_1=m_2=\cdots=m_t=m, we give the necessary and sufficient condition that there exist orthogonal matrices P of order m and Q of order n such that A_i=P\left(\beginarrayllC_i & 0 \\ 0 & 0\endarray\right) Q^\prime, i =1, \cdots, t . B_j=Q \Lambda_j Q^\prime, j=1, \cdots, I, where C_1, \cdots, C_t, \Lambda_1, \cdots, \Lambda_I are all diagonal matrices. For the case of m_i \neq m; the necessary and sufficient condition is derived too. We also obtain the necessary and sufficient condition for simultaneous diagonalization of B_1, \cdots, B I by pre- and post- multiplication by C^\prime and C, where C is nonsingalar.