Positive definiteness of a function \(\mathbb{R}^{d} \times \mathbb{R}^{d} ightarrow \mathbb{R}\) means that, for integer \(n \geq 1\) and each finite collection of points \(\mathbf{x}=\left\{x_{1}, \ldots, x_{n}ight\}\), the \(n \times n\) matrix \(K[\mathbf{x}, \mathbf{x}]\) with components

\[
K_{i j}=K\left(x_{i}, x_{j}ight)
\]

is positive definite and symmetric.

Let \(x\) be a point in real Euclidean space \(\mathbb{R}^{d}\). For each \(\lambda>0\), the function

\[
K\left(x, x^{\prime}ight)=e^{-\lambda\left\|x-x^{\prime}ight\|}
\]

is the covariance function for a stationary autoregressive process of order one if \(d=1\), also called the Ornstein-Uhlenbeck process for general \(d\). Show that \(K\) is positive definite.