§ 67] MINIMUM EQUATION OF MATRIX 109 
g(w)M(m — ü:I)=f(œ)I^g{œ)q{œ)I. 
67. Minimum equation of a matrix. Any square 
matrix m with elements in a field F is a root of its char 
acteristic equation (§ 59) and hence is a root of a unique 
equation </>(co)=o of lowest degree whose coefficients 
belong to F, the leading coefficient being unity. This 
equation is called the minimum (or reduced) equation 
of m. It is understood that when co is replaced by m, 
the constant term of 4>{co) is multiplied by the unit 
matrix /. 
Lemma. If \(m)=o, where X(co) is a polynomial 
with coefficients in F, then X(co) is exactly divisible by <p{co). 
For, let q{co) and r{00) denote the quotient and re 
mainder from the division of X(co) by </>(co), where r{ co) 
is either zero identically or is of degree less than that of 
<£(co). Then 
Xico) = ç(’co)0(co)+r(co). 
Hence r{m) = o , so that r(co) is zero identically. 
Theorem i. The minimum equation of an n-rowed 
square matrix m is qffi) —o, where q{of) is the quotient of 
the characteristic determinant /(co) of m by the greatest 
common divisor g(co) of its (n — i)-rowed minors. 
Denote the adjoint matrix (§3) of m — ixl by 
{m— co7) 0 . Each of its elements is divisible by g(co). 
Hence 
(m—œI) 0 =g{w)M, 
where M is a matrix whose elements are polynomials in 
co without a common factor other than a number of F. 
Hence (16) with x — m becomes