We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. Let $\alpha \in \mathbf{R}^p$ (resp. $a \in \mathbf{R}^p$) be the vector with coordinates $(\alpha_i)_{i \in \llbracket 1,p \rrbracket}$ (resp. $(a_i)_{i \in \llbracket 1,p \rrbracket}$) and $h_\alpha = \sum_{i=1}^p \alpha_i \tau_{x_i}(\gamma_{2\lambda})$. The matrix $K$ is defined by $K_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2\lambda}\right)$ (here in the case $d=1$). (a) Show that $h_\alpha$ is an interpolant if and only if $K\alpha = a$ where $K$ is the matrix introduced in question (6) (here in the case $d = 1$). (b) Show that $K$ is invertible.
We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. Let $\alpha \in \mathbf{R}^p$ (resp. $a \in \mathbf{R}^p$) be the vector with coordinates $(\alpha_i)_{i \in \llbracket 1,p \rrbracket}$ (resp. $(a_i)_{i \in \llbracket 1,p \rrbracket}$) and $h_\alpha = \sum_{i=1}^p \alpha_i \tau_{x_i}(\gamma_{2\lambda})$. The matrix $K$ is defined by $K_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2\lambda}\right)$ (here in the case $d=1$).
(a) Show that $h_\alpha$ is an interpolant if and only if $K\alpha = a$ where $K$ is the matrix introduced in question (6) (here in the case $d = 1$).
(b) Show that $K$ is invertible.