OF ARTS AND SCIENCES. 217 



6. Let $ be any real improper orthogonal matrix. Then if w 

 denotes a matrix whose determinant is equal to —1, and whose con- 

 stituents are all zero except those in the princi^jal diagonal which are 

 severally equal to ±1, the matrix 



is a real proper orthogonal matrix. For 



= 3>$ 

 = 1. 



Moreover, 



|<^| = |w|.|<J'| = l. 



Therefore, we may put 



^ Vl + Y/ 



for a proper choice of the real skew symmetric matrix Y. Whence, 

 since w^ = 1, we derive 



/I - Y\2 



