4 1. POLYNOMIAL S I N ON E VARIABL E
0282
0282
0177
0177
0174
0.06871
0.06871
0.0190i
0.01901
0.11841
0.11841
-1.0174 -
ETC...ETC..
We not e tha t convenien t facilitie s ar e availabl e fo r callin g ma t lab insid e o f mapl e
and fo r callin g mapl e insid e o f matlab . W e encourag e ou r reader s t o experimen t
with th e passag e o f data betwee n thes e tw o programs .
Some numerical methods for solvin g a univariate polynomial equation p(x) = 0
work b y reducing this problem t o computing th e eigenvalue s of the companio n ma -
trix ofp(x), whic h is defined a s follows. Le t V denot e the quotient o f the polynomia l
ring modul o th e idea l (p(x)) generate d b y the polynomia l p{x). Th e resultin g quo -
tient rin g V = Q[x]/(p(x)) i s a d-dimensiona l Q-vecto r space . Multiplicatio n b y
the variabl e x define s a linear ma p fro m thi s vecto r spac e t o itself .
(1.2) Tiniest V - V,f(x) ^x-f(x).
The companion matrix i s th e d x d-matri x whic h represent s th e endomorphis m
Times^ wit h respec t t o th e distinguishe d monomia l basi s {l,x,x
2
,
V. Explicitly , th e companio n matri x o f p(x) look s lik e this:
/ 0 0 0 -a
0
/ad \
1 0 0 —ai/ad
r.d-1
} o f
(1.3) TimesT
0 1
V 0 0
0
-ail^d
1 -a d-i/ad )
PROPOSITION
1.2. The zeros ofp(x) are the eigenvalues of the matrix Times x.
PROOF.
Suppos e tha t f(x) i s a polynomia l i n C[x] whose imag e i n V 0 C =
C[x]/(p(x)) i s an eigenvector o f (1.2) wit h eigenvalue A . The n x- f(x) \- f(x) i n
the quotien t ring , which mean s that {x A) f(x) i s a multiple oip(x). Sinc e f(x)
is not a multiple of p(x), w e conclude that A is a root otp(x) a s desired. Conversely ,
if ii i s an y roo t o f p(x) the n th e polynomia l f(x) = p(x)/(x p) represent s a n
eigenvector o f (1.2) wit h eigenvalu e \i.
COROLLARY
1.3. The following statements about p(x) G Q[x] are equivalent:
The polynomial p{x) is square-free, i.e., it has no multiple roots in C .
The companion matrix Times
x
is diagonalizable.
The ideal (p(x)) is a radical ideal in Q[x].
We not e tha t th e se t o f multipl e root s o f p(x) ca n b e compute d symbolicall y
by formin g th e greates t commo n diviso r o f p(x) an d it s derivative :
(1.4) q(x) gcd(p(x),p\x))
Thus th e thre e condition s i n th e Corollar y ar e equivalen t t o q(x) = 1.
Every ideal in the univariate polynomial ring Q[x] is principal. Writin g p(x) fo r
the idea l generato r an d computin g q(x) fro m p(x) a s i n (1.4), we get th e followin g
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