# Heat kernel derivation from heat equation

While reading this paper, I couldn't understand how they derived heat kernel $H_t$ from the heat equation,

$$\frac{\delta{u}_t}{\delta{t}} = - \mathcal{L}u_t$$

When I take the integration, I can derive the equation of heat $u(t)$ at every node but I am unable to get the heat kernel. I would be very thankful if someone could give an overview explanation or some pointers as to how they got from equation (1) to (3).

Write $u_t=\sum a_j(t) \phi_j$. Then the PDE becomes (using $\phi_j$ are eigenfunctions) $$\sum a_j'(t) \phi_j = -\sum \lambda_j a_j(t) \phi_j.$$ As $\phi_j$ are a basis, $a_j'(t)=-\lambda_j a_j$ and so solving the ODE gives $a_j(t)=e^{-\lambda_j}a_j(0)$. With $a_j(0)=\phi_j^T u_0$ you arrive at the expression (2).