grandes-ecoles 2023 Q12

grandes-ecoles · France · mines-ponts-maths2__pc Matrices Matrix Algebra and Product Properties

For $X, Y \in \mathscr{M}_{N,1}(\mathbf{R})^2$, we define $$\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$$ where $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ is a probability with $\pi[j] \neq 0$ for all $j$. Show that $(X, Y) \mapsto \langle X, Y \rangle$ is an inner product on $\mathscr{M}_{N,1}(\mathbf{R})$.

For $X, Y \in \mathscr{M}_{N,1}(\mathbf{R})^2$, we define
$$\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$$
where $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ is a probability with $\pi[j] \neq 0$ for all $j$.\\
Show that $(X, Y) \mapsto \langle X, Y \rangle$ is an inner product on $\mathscr{M}_{N,1}(\mathbf{R})$.

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