東京大学 新領域創成科学研究科 メディカル情報生命専攻 2023年8月実施 問題8
Author
zephyr
Description
Answer the following questions regarding the transition matrix
,
.
(1) Show that the vector
is an eigenvector of .
(2) Find the -th order transition matrix .
(3) Find .
The trace of an real-valued matrix is defined as . Answer the following questions.
(4) Prove that where the eigenvalues of are described as .
(5) Prove the following: If there exists a natural number such that ( is the zero matrix), then .
回答以下关于转移矩阵
\begin{aligned}
\mathbf{P}\begin{pmatrix} p \ -q \end{pmatrix} &=
\begin{pmatrix}
1 - p & p \
q & 1 - q
\end{pmatrix}\begin{pmatrix} p \ -q \end{pmatrix} \
&= \begin{pmatrix}
(1-p)p + p(-q) \
qp + (1-q)(-q)
\end{pmatrix} \
&= \begin{pmatrix}
p - p^2 - pq \
qp - q + q^2
\end{pmatrix} \
&= \begin{pmatrix}
p(1 - p - q) \
-q(1 - p - q)
\end{pmatrix} \
&= (1 - p - q)\begin{pmatrix} p \ -q \end{pmatrix}
\end{aligned}
\mathbf{S} = \begin{pmatrix} 1 & p \ 1 & -q \end{pmatrix}, \quad
\mathbf{\Lambda} = \begin{pmatrix} 1 & 0 \ 0 & 1-p-q \end{pmatrix}
\mathbf{P}^n = \mathbf{S}\mathbf{\Lambda}^n\mathbf{S}^{-1} =
\frac{-1}{p + q}\begin{pmatrix} 1 & p \ 1 & -q \end{pmatrix}
\begin{pmatrix} 1 & 0 \ 0 & (1-p-q)^n \end{pmatrix}
\begin{pmatrix} -q & -p \ -1 & 1 \end{pmatrix}
\lim_{n \to \infty} \mathbf{P}^n = \begin{pmatrix}
\frac{q}{p+q} & \frac{p}{p+q} \
\frac{q}{p+q} & \frac{p}{p+q}
\end{pmatrix}
\begin{aligned}
\mathrm{tr}, \mathbf{A} &= \mathrm{tr}, (\mathbf{P}\mathbf{D}\mathbf{P}^{-1}) \
&= \mathrm{tr}, (\mathbf{D}\mathbf{P}^{-1}\mathbf{P}) \quad \text{(cyclic property of trace)} \
&= \mathrm{tr}, \mathbf{D} \
&= \sum_{i=1}^{n} \lambda_i
\end{aligned}