TABER. — LINEAR TRANSFORMATIONS. 381 



(fi 1 -Y)- 1 (fi 1 + Y), 



in which the symmetric matrix Y is subject to the condition that 

 f} "~ 2 Y ft ' * is commutative with fy, - *!^ i) ~*, and is such that the 

 determinant of | Q — Y | ^ 0. 



To find the most general skew symmetric matrix O - iYfi ~i 

 commutative with Q~*QiQ ~s requires in general the solution of an 

 algebraic equation of the nth degree.* 



Again, since i{/ is the product of three orthogonal matrices of which 

 — 1 is not a latent root and which are commutative with Cl 1} the group 

 of solutions of the equation $ Q (/> = fi is a sub-group of the group of 

 solutions of the equation $ O <£ = fi generated by the totality of 

 expressions 



n-i (i - y)- 1 (i + y) n i = (Oo - iy Y<y)-» (Oo + no*Y$V), 



in which Y is a skew symmetric matrix commutative with fi x and such 

 that the determinant of Y — 1 is not zero. 



The sub-group consisting of the orthogonal solutions of the equa- 

 tion <p£}(p = Q, is the group of orthogonal matrices commutative with 

 fi and Qi. It has been shown that the group of orthogonal matrices 

 commutative with Qi is generated by the totality of expressions 

 (1 — Y) -1 (1 -f Y) in which Y is a skew symmetric matrix commu- 

 tative with fii. This group contains only proper orthogonal matrices; 

 and it is readily proved that the group of orthogonal matrices commu- 

 tative with Q is generated by the totality of expressions (1 — Y) _1 

 (1-f Y) in which Y is commutative with O and such that|Y — 1 |4=0. 

 Therefore the group of orthogonal solutions of the equation Q<p = 12 

 can be generated by the totality of orthogonal matrices of which —I 

 is not a latent root that are commutative with Q ; that is, by the total- 

 ity of expressions (1 — Y) -1 (1 + Y), in which Y is a skew symmetric 

 matrix commutative with Q and such that J Y — 1 | ^= 0. 



Clark University, Worcester, Mass. 



* If the fundamental syzygy in ft — - n. 1 ft,, - 2 is of order equal to the order 

 of this matrix, the most general matrix commutative with ft -5 ft 1 ft — 2 is 

 a rational integral function of this matrix. This is the case if the roots of 

 the equation I ft — //ft I = are all distinct. The fundamental syzygy in any 

 matrix <p is the polynomial in powers of <p of lowest order that vanishes. 



