On my lecture notes, I came across the statement: 'Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix.'
Can someone explain why this is the case?
The context behind this question is finite element method.
The claim that "Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix" is more of a slogan than a precise theorem. This answer gives the usual surface level explanation without going into the technical details.
To explain the observation that lies behind this slogan lets take a regular and quasi-uniform mesh of the computational domain that is characterized by $h > 0$ being a diameter of a smallest mesh element. Let $\kappa(A)$ denote the condition number of a matrix $A$ computed as $\kappa(A) = \frac{\lambda_{max}}{\lambda_{min}}$ where $\lambda_{max}$ and $\lambda_{min}$ are the highest and lowest eigenvalues of $A$. The lower the condition number is the 'easier' is to solve the equation with the matrix.
Let $M$ denote the mass matrix and $K$ denote the stiffness matrix. It can be shown that $\kappa(M) \sim 1$ with respect to $h$ and $\kappa(K) \sim h^{-2}$. So $h \to 0$ leads to $\kappa(A) \to \infty$ which supports the slogans statement.
Another way in which it is 'easier' to invert the mass matrix than stiffness matrix is through preconditioned Krylov space method. For mass matrix even a diagonal of the matrix can serve as a suitable preconditioner whereas constructing an efficient preconditioner for the stiffness matrix is much more convoluted.