MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 7 



We can think of this as a special multiplication of hyperquantities a, b 

 and s. If we define 



Cik = aijbjk (4.2) 



we may go from the y's to the z's directly thru yi = CikZk- We can now 

 consider the "table" 



'aiiCi2 . . . c„^ 



\aml . ' . amn 



as being the quantity a, and the table 



/bnbn .... 



\"pl Oprnf 



as the quantity b. 



These "tables" are called matrices. 



Going to eq. (4.2) we see that the quantity c is to be a "table," the typical 

 element dj of which is to be gotten by multiplying the tth row of a by the 

 yth column of 6, term by term thus: 



C,y = Giibij + diibij + . . . 



After a little practice it becomes almost automatic to form the y th term 

 of the product of two matrices by letting the index finger of the left hand 

 follow across the ith row of the left matrix while the right index finger 

 follows down theyth column of the right matrix. The fingers step along in 

 synchronism and at each pause the quantities under the two fingers are 

 multiplied and the product added algebraically to the accumulated sum. 



The algebra of these special multiplications is not commutable, i.e. 

 ab 9^ ba. 



Eq. (4.1) can be considered as a special case of eq. (4.2), in which the 

 matrices x and y have one column only. In this manner a vector with 



. h\ 



components xi xo xs can be considered as the matrix I .T2 I . 



If eq. (4.1) has the same number of .r's as y's we may solve (by means of 

 determinants) for the x's in terms of the y's. We would then get a new set 

 of equations 



Xi = a~j y,: 



