# Why is easier to get inverse of mass matrix?

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.

• You're going to have to provide more context. What kind of equation are you solving? Can you give an example where it's easier to invert the mass matrix than to solve a system with the stiffness matrix? Feb 25 at 6:36
• @K.Jiang The context is sort of given by the tag 'finite-element-method'. Regardless to the equation that is being solved, 'mass matrix' usually means matrix that corresponds to the $L^2$ scalar product, 'stiffness matrix' corresponds to the form that involves 1st derivatives like in linear elasticity or Laplacian operator. This is common jargon of FEM.
– Korf
Feb 25 at 12:43
• This claim makes not much sense (It seems to imply that the inverse of the stiffness matrix might be computed?). The mass matrix is a structured (at least for mostly regular grids) sparse matrix, its inverse is dense. It is more economic to pre-compute a sparse decomposition, like LU or QR, and use that to solve the linear system of the step equation. Feb 25 at 16:32

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.