例 7.9

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• 例 7.10
• 例 7.15

例 7.9

设 $f(\lambda ),g(\lambda )$ 是数域 $K$ 上的首一多项式， $d(\lambda )=(f(\lambda ),g(\lambda ))$，$m(\lambda )=[f(\lambda ),g(\lambda )]$ 分别是 $f(\lambda )$ 和 $g(\lambda )$ 的最大公因式和最小公倍式，证明下列 $\lambda$-矩阵相抵：

$$\left(\begin{array}{cc}f(\lambda )& 0\\ 0& g(\lambda )\end{array}\right),\left(\begin{array}{cc}g(\lambda )& 0\\ 0& f(\lambda )\end{array}\right),\left(\begin{array}{cc}d(\lambda )& 0\\ 0& m(\lambda )\end{array}\right).$$

解答

证明 由已知，存在多项式 $u(\lambda ),v(\lambda )$，使得 $f(\lambda )u(\lambda )+g(\lambda )v(\lambda )=d(\lambda )$。设 $f(\lambda )=d(\lambda )h(\lambda )$，则 $m(\lambda )=g(\lambda )h(\lambda )$。作下列 $\lambda$-矩阵的初等变换：

$$\begin{array}{rl}\left(\begin{array}{cc}f(\lambda )& 0\\ 0& g(\lambda )\end{array}\right)& \to \left(\begin{array}{cc}f(\lambda )& 0\\ f(\lambda )u(\lambda )& g(\lambda )\end{array}\right)\\ & \to \left(\begin{array}{cc}f(\lambda )& 0\\ f(\lambda )u(\lambda )+g(\lambda )v(\lambda )& g(\lambda )\end{array}\right)\\ & =\left(\begin{array}{cc}f(\lambda )& 0\\ d(\lambda )& g(\lambda )\end{array}\right)\to \left(\begin{array}{cc}0& -g(\lambda )h(\lambda )\\ d(\lambda )& g(\lambda )\end{array}\right)\\ & \to \left(\begin{array}{cc}0& g(\lambda )h(\lambda )\\ d(\lambda )& 0\end{array}\right)\to \left(\begin{array}{cc}d(\lambda )& 0\\ 0& m(\lambda )\end{array}\right).\end{array}$$

另一结论同理可得。$\square$