no CHARACTERISTIC, RANK EQUATIONS [chap, vn 
We may delete the common factor g(co) from this identity 
in matrices since it is equivalent to n 2 equations between 
elements of the w-rowed matrices. Thus 
(28) M{m—uI) = q{w)I. 
As in § 59 this identity holds true after co is replaced by 
any matrix commutative with m, say m itself. Hence 
q(m) = o. By the lemma, g(co) is divisible by 0(co), 
If p is another indeterminate, we have 
p)(p —co), 
where \f/(co, p) is a polynomial in co and p with coefficients 
in F. We may replace p by m and, since 4>(m) = o, 
obtain 
0(co)/ = i/'(co, m)(m — wT) . 
From this and (28), we deduce 
q{u>)\p{w, — = . 
We may delete the common factor m — co7 whose deter 
minant is not zero identically in co. Since the elements 
of M have no common factor, q(co) must divide </>(co). 
Our two results show that </(co) and 0(co) differ only 
by a factor belonging to the field F. Hence the theorem 
is proved. 
Theorem 2. Every root of the characteristic equation 
/(co) =0 of a matrix is a root of its minimum equation 
0(co) =0, and conversely. 
For, if we pass from (28) to determinants, we have 
I M | */(co) = [0(co)]”. 
The converse is true by Theorem 1.