Greatest common divisor of polynomials

  • #1
HI there,

I have a tiny question concerning the gcd of polynomials. Assume, [itex]\chi[/itex] is the greatest common divisor of the polynomails [itex]p_{ij}, i,j=1,2[/itex]. I then form
[tex]
q_{11}=p_{11}^2+p_{12}^2,\quadd q_{12}=p_{11}p_{21}+p_{12}p_{22},\quadd q_{21}=p_{21}p_{11}+p_{22}p_{12},\quadd q_{22}=p_{21}^2+p_{22}^2
[/tex]

which is in fact a matrix multiplication:
[tex]
\left(\begin{array}{cc}
q_{11} & q_{12} \\ q_{21} & q_{22}
\end{array}\right) = \left(\begin{array}{cc}
p_{11} & p_{12} \\ p_{21} & p_{22}
\end{array}\right)
\left(\begin{array}{cc}
p_{11} & p_{12} \\ p_{21} & p_{22}
\end{array}\right)^T
[/tex].

I think everyone would agree that [tex]\chi^2[/itex] is a common divisor of all the q's. But is it also the greatest one...? I couldn't find a counterexample and couldn't prove it either.

Thanks a lot :smile:
 
Last edited:

Answers and Replies

  • #2
14,131
11,418
##2 \,|\,\begin{bmatrix}26&36\\36&50\end{bmatrix} = \begin{bmatrix}1&5\\1&7\end{bmatrix} \cdot \begin{bmatrix}1&1\\5&7\end{bmatrix}=\begin{bmatrix}1&5\\1&7\end{bmatrix} \cdot \begin{bmatrix}1&5\\1&7\end{bmatrix}^\tau##
 

Related Threads on Greatest common divisor of polynomials

Replies
45
Views
11K
  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
1
Views
2K
Replies
1
Views
3K
Replies
4
Views
8K
Replies
17
Views
6K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
Top