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erlang

P8 IMC 2017

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Definisikan barisan matriks $A_1,A_2,...$ dengan relasi rekursi berikut: $A_1=\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$, $A_{n+1}=\begin{bmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n\end{bmatrix}$ untuk $n\ge 1$ dimana $I_m$ adalah matrix identitas $m\times m$. Buktikan $A_n$ punya $n+1$ eigenvalue bilangan asli berbeda $\lambda_0<\lambda_1<....<\lambda_n$ dengan multiplisitas ${n\choose 0},{n\choose 1},...,{n\choose n}$ secara berturut-turut.

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Misalkan $P_n(t)$ adalah characteristic polynomial $A_n$. Perhatikan kalau $P_{n+1}(t)=det(tI-A_{n+1})=det((tI-A_n)^2-I)=det(tI-A_n-I)det(tI-A_n+I)=P_n(t+1)P_n(t-1)$. Jelas $P_1=(t+1)(t-1)$. Lalu mudah di cek dari relasi rekursinya kalau untuk setiap $n$ asli, $\lambda_i=-n+2i$ dengan multiplisitas ${n\choose i}$ (relasinya sama dengan paskal)

 

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