166 
PROF. H. F. BAKER ON CERTAIN LINEAR 
For ( b ), since 
PO, (u) p- 1 = 0 (pup- 1 ), trs (p- 1 A) = — A p~\ 
we have the following transformations of matrices 
[1} (/VIA)] -1 = trs [Q (A/3 -1 )] = trs [/3Q (p- 1 A) /3 -1 ] = /V 1 [trs Q (/3 -1 A)] /3, 
and hence, writing LL 1 (u) for [1} (w)] -1 , the following equations among determinants 
| Q -1 (p- 1 A)- P | = | trs Q (p- l A)- P | = | Q (p- 1 A)- P \, 
which establishes the result in question. 
§ 16. In many dynamical applications the matrix A is a sum of two matrices 
A = a + A 
where a is a symmetrical matrix of real constants, and L a symmetrical matrix whose 
elements are small. Suppose, further, that p, denoting a row of 2 n real variables 
p x , p 2 , ..., the matrix a is such that the quadratic form 
— — ^ ij P i P] 
does not vanish unless every one of the 2 n elements of p is zero, which requires that 
the determinant \a\ is not zero. Then, if this quadratic form be denoted by ap 2 , 
and if each of p and >j be a row of 2 n real quantities, the form 
a.(g+ir]) (g.— m), = a.g 2 + ia (]£—&)■+arf, = a(g 2 + p), 
has the same property. 
When this is so, it can be shown that the roots of the determinantal equation in \}s, 
j p- y a — A | = 0, 
are pure imaginaries, and that the invariant factors of the matrix /3 _1 a — \Jy are linear. 
As the proof is not long it may be given here (cf. ‘ Proc. Lond. Math. Soc.,’ XXXV., 
December 11, 1902, p. 380). 
Let \Js satisfy the determinantal equation 
a—P\Jj |=0; 
as the determinant \a \ is not zero, \fs cannot be zero. Then 2 n quantities x x , x 2 , ..., 
whose aggregate is denoted by x, can be taken to satisfy the 2 n linear equations 
(a— P\}s) x = 0. 
If x 0 denote the row formed by the 2 n quantities which are the conjugate complexes 
of those of x, we have in turn 
aXXy = \jsPxX 0 , oixpc — ’xJ/'PXqX, olXqX — —yJrpX^X, 
