grandes-ecoles 2023 Q11

grandes-ecoles · France · mines-ponts-maths2__pc Matrices Eigenvalue and Characteristic Polynomial Analysis

We consider a Markov kernel $K$. We assume that 1 is a simple eigenvalue of $K$. We assume that there exists a probability $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ such that:
(a) For all $j \in \llbracket 1;N \rrbracket$, $\pi[j] \neq 0$.
(b) $\forall (i,j) \in \llbracket 1;N \rrbracket^2$, $\pi[i] K[i,j] = K[j,i] \pi[j]$; we say that $K$ is $\pi$-reversible. Show that $\pi K = \pi$.

We consider a Markov kernel $K$. We assume that 1 is a simple eigenvalue of $K$. We assume that there exists a probability $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ such that:\\
(a) For all $j \in \llbracket 1;N \rrbracket$, $\pi[j] \neq 0$.\\
(b) $\forall (i,j) \in \llbracket 1;N \rrbracket^2$, $\pi[i] K[i,j] = K[j,i] \pi[j]$; we say that $K$ is $\pi$-reversible.\\
Show that $\pi K = \pi$.

Paper Questions

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