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LI. Proof of Professor Sylvester's " Third Law of Motion." 

 By Akthub Buchheim, M.A* 



THE " third law of motion " in the theory of matrices is 

 that " the nullity of the product of two matrices is not 

 less than the greater of their nullities, but not greater than 

 the sum of the two nullities." Prof. Sylvester has given an 

 outline of a proof of this theorem (which he regards as 

 " perhaps the most important in the whole theory of multiple 

 quantity") in his lecture "On the three Laws of Motion in 

 the World of Universal Algebra;" f but I have his authority 

 for stating that he considers this proof as wanting in sim- 

 plicity, so far as the superior limit is concerned. The instan- 

 taneous proof here given is merely an extension of the proof 

 given by Clifford for ternary matrices. 



A linear space of (a — 1) dimensions will be called an 

 «-point : the coordinates of any point of an a-point are linear 

 functions of the coordinates of a points determining it. A ma- 

 trix A of order w-is considered as operating on the coordinates 

 of the points of an w-point. Calling the coordinates («^ 1 ...^ ra ), 

 there are in general n points satisfying the equations A(^ x ...#») 

 =\(%i . . . #„), where X is a scalar : X is one of the latent roots 

 of A, and the point (<% . . . oc„) is called the latent point apper- 

 taining to \. It follows from known theorems on the solution 

 of linear equations, that if A is of nullity a, the equation 

 A(.^ . . . x„) = is satisfied by all the points of a certain a-point, 

 and conversely. I call this a-point the null space of A, and I 

 denote it by R : any point of R may be said to be destroyed 

 by A. There is also an (n— a)-point S, which I call the space 

 preserved by A : all the points of the w-point which are not 

 destroyed are transformed into points of S : S is, in fact, the 

 (w— a)-point determined by the latent points appertaining to 

 the latent roots that do not vanish. 



Now let B be a matrix of nullity /3 ; let W be its null space, 

 and S' the space preserved by it : then I prove that, if R, S' 

 intersect in a S-point (T), the nullity of AB is /3 + 8. 



To find the nullity we have only to find the most general 

 space (the space of most dimensions) that B transforms into 

 T ; for this is obviously the most general space destroyed by 

 AB. Now let U be the S-point in S which B transforms 

 into T ; then it is easy to see that the most general space 

 which B transforms into T will be the (S + /3) -point joining 

 (U, B/). From this it follows that the most general space 

 destroyed by AB is a (8 + /3)-point — that is, that the nullity 



* Communicated by the Author. 

 t 3 Johns Hopkins Circulars, 33. 

 2H2 



