A.5.3 3$B=EBP3Q9TNs$N8GM-B?9`<0$H(B Strum $BNs(B

$$T=\left( \begin{array}{ccccc} a_1 & b_1 & 0 & & \bigzerou \\ b_1... ...-2} & a_{N-1} & b_{N-1} \\ \bigzerol & & 0 & b_{N-1} & a_N \end{array}\right)$$

$B$rN;0=EBP3Q9TNs$H$9$k!#(B

$b_k\ne 0$ ( $k=1,2,\cdots,N-1$) $B$H2>Dj$9$k!#$b$7$"$k(B $k$ $B$KBP$7$F(B $b_k\ne 0$ $B$J$i$P(B

$$T = \left(\begin{array}{cc}T' & O \\ O & T''\end{array}\right)$$

$B$H$J$j!"(B$T'$, $T''$ $B$N8GM-CM$r5a$a$kLdBj$K5"Ce$G$-$k$+$i!"0lHL@-$O<:(B $B$o$l$J$$!#(B

$p_k(\lambda)$ $B$r(B $\lambda I-T$ $B$NBh(B $k$ $B $k=0,1,\cdots,N$)$B!#$9$J$o$A(B

$$p_k(\lambda):= \left\{ \begin{array}{ll} \det(\lambda I_k-T_k) & \mbox{($k=1,2,\cdots,N$)}\\ 1 &\mbox{($k=0$)} \end{array} \right. .$$ (A.7)

$B$?$@$7!"(B

$$I_k=$$$$\mbox{$k$\ $B<!$NC10L9TNs(B}$$$$,\quad T_k=\left( \begin{array}{ccccc} a_1 & b_1 & 0 & & \bigzero... ...2} & a_{k-1} & b_{k-1} \\ \bigzerol & & 0 & b_{k-1} & a_k \end{array}\right).$$

$B$9$0J,$+$kL?Bj$rFs$D!#(B

Proof. $B9TNs<0$N9T$K4X$9$kE83+DjM}$rMQ$$$k!#(B$\qedsymbol$ ARRAY(0x12f6258) $\qedsymbol$

Proof.

1. $B$b$7$b(B $p_k(\lambda_0)=p_{k+1}(\lambda_0)=0$ $B$H$9$k$H!"A22=<0$+$i(B ${b_k}^2p_{k-1}(\lambda_0)=0$. $b_k=0$ $B$H2>Dj$7$?$+$i(B $p_{k-1}(\lambda_0)=0$. $B$3$l$r7+$jJV$9$H(B
   $$0=p_{k+1}(\lambda_0)=p_k(\lambda_0)=p_{k-1}(\lambda_0)=\cdots =p_{2}(\lambda_0)=p_1(\lambda_0)=p_0(\lambda_0).$$
   $B$3$l$+$i(B
   $$p_0(\lambda_0)=0$$
   $B$3$l$O(B $p_0\equiv 1$ $B$KL7=b$9$k!#(B
2. $p_k(\lambda_0)=0$ $B$rA22=<0$KBeF~$9$k$H(B $p_{k+1}(\lambda_0)=-{b_k}^2 p_{k-1}(\lambda_0)$. $BA09`$h$j:8JU(B $\ne 0$. $B$3$l$+$i(B $p_{k+1}(\lambda_0)$, $p_{k-1}(\lambda_0)$ $B$O0[Id9f$G$"$k!#(B $\qedsymbol$
3. $p_0(\lambda)\equiv 1$ $B$G$"$k$+$iL@$i$+!#(B
4. $BA22=<0(B

   $$p_{k}(\lambda)= (\lambda-a_{k})p_{k-1}(\lambda)-{b_{k-1}}^2 p_{k-2}(\lambda)$$ (A.8)

   
   
   $B$rHyJ,$9$k$H!"(B

   $$p_{k}'(\lambda)= p_{k-1}(\lambda)+(\lambda-a_{k})p_{k-1}'(\lambda) -{b_{k-1}}^2p_{k-2}'(\lambda).$$ (A.9)

   
   

   (A.8) $B$H(B (D.5) $B$+$i(B

   $$p_{k}'(\lambda)p_{k-1}(\lambda)-p_{k}(\lambda)p_{k-1}'(\lambda) =... ...)p_{k-2}(\lambda)-p_{k-1}(\lambda)p_{k-2}'(\lambda) \right) +p_{k-1}^2(\lambda)$$ (A.10)

   
   
   $B$,F@$i$l$k!#$3$3$G(B
   $$q_{k}(\lambda) := p_{k}'(\lambda)p_{k-1}'(\lambda) -p_{k}(\lambda)p_{k-1}(\lambda)$$
   $B$H$*$/$H(B (A.10) $B$O(B
   $$q_{k}(\lambda)=p_{k-1}(\lambda)^2 +\beta_{k-1}(\lambda)^2q_{k-1}(\lambda)$$   $$\mbox{($k=2,3,\cdots,N$)}$$$$.$$
   $B$H$J$k!#$H$3$m$G(B
   $$q_1(\lambda)=p_{1}'(\lambda)p_{0}(\lambda) -p_{1}(\lambda)p_{0}'(\lambda) =p_{1}'(\lambda)=1\cdot1-(\lambda-\alpha_1)\cdot 0=1>0$$
   $B$G$"$k$+$i!"0J2<5"G0\quad\mbox{($k=2,3,\cdots,N$)}. \end{displaymath} -->
   $$q_k(\lambda)>0$$   $$\mbox{($k=2,3,\cdots,N$)}$$$$.$$
   $BFC$K(B
   $$q_{N}(\lambda)=p_{N}'(\lambda)p_{N-1}(\lambda) -p_{N}(\lambda)p_{N-1}'(\lambda)>0$$
   $B$G$"$k$,!"(B $p_N(\lambda)=0$ $B$G$"$k$+$i(B
   $$p_{N}'(\lambda)p_{N-1}(\lambda)>0. \quad\qed$$

ARRAY(0x12f648c) $\qedsymbol$