Information Content of Tracer Data With Respect to Steady-state Systems 183 



(c) The n a,- and rr" Aj.j comprise a necessary and sufficient set of data to 

 define uniquely the parameters of the system. 



(d) A simple matrix relation (3) exists between the Aj^^ and a,- of the data and 

 the A,y of the system. This relation can be written: 



Ml = \A l«l 



or 



where 





I3I 



— Aj2 — Aj3 



/122 — "^ 



23 



1 



32 



a = 



h 



33 



ai 



(-11 



(4) 

 (5) 



A, 



11 

 1 



'31 



All 

 -^22 



^32 



'13 

 '23 

 ^33 





 





 iy.2 

 0" a. 



Equation (5) expresses the system parameters in terms of the invariants in 

 the data. If these invariants are known, the fractional turnover rates, Aj-;, 

 can all be determined. However, in most cases the experimental data are 

 incomplete in that certain of the A^j and a^ are not known. For these cases, 

 an infinity of models mathematically consistent with the data can be obtained 

 from equation (5) by inserting arbitrary values for the unknown Aj.j and a^, 

 preserving the initial conditions and other constraints in the data. Most of 

 these arbitrary models, however, will be physically meaningless because some of 

 the fractional turnover rates will be negative. Consequently, it is necessary to 

 investigate what range of values of the unknown A^^ and a, correspond to 

 physically meaningful models. This can be done by relating variations in A^^j 

 and a.; to variations in the X^j. 



One may define (2) a matrix |P| in such a way that the product \PA\ will 

 preserve the known A^j. The number of variables in \P\ will be equal to the 

 degrees of freedom in the Aj^j. If both sides of equation (4) are premultiplied by 

 the matrix \P\ this equation can be rewritten: 



(6) 

 (7) 



a 



\PXP-'^\ \PA\ = \PA\ 

 which is of the form 



[A'l l^'l = |/1'| |a| 

 where 



M'l = l^^l (8) 



\l'\ = \PKP-^\ (9) 



Equation (9) expresses a mapping of the matrix \X\ corresponding to varia- 

 tions in the unknown Aj,j only. It also represents a general solution of all 

 models mathematically consistent with the data in terms of a minimum number 

 of variables. This solution is expressed in terms of an arbitrary model represented 

 by the matrix \X\. 



Similarly, we can define a matrix \D\ so that the product |aZ)| will preserve 

 all the known a^. Incorporating this into equation (4), we get 



|;.^Z)^-i||^| = |y4||aZ)| (10) 



