# Why the product of symmetric-sparse matrices is not symmetric, or dense

For the equation $\frac{\partial u}{\partial t}=-(-\Delta)u$ with zero boundary condition using finite element method. Applying the corresponding weak formulation and taking $v=\phi_{j},j=1,2,...,N$, We obtain the following equation.

$$\sum\limits_{i=1}^{N} \frac{du_{i}}{dt}\int _{\Omega} \phi_{i}\phi_{j}d\Omega=-\sum\limits_{i=1}^{N}u_{i} \int _{\Omega} \nabla\phi_{i}.\nabla\phi_{j}d\Omega\,.$$

Introducing the mass $\mathcal {M}_{i,j}=\int _{\Omega} \phi_{i}\phi_{j}d\Omega$ and stiffness matrices $\mathcal{K}_{i,j}=\int _{\Omega} \nabla\phi_{i}.\nabla\phi_{j}d\Omega$.

Then we have $$\frac{d\bf{u}}{dt}=-\mathcal{M}^{-1}\mathcal{K}\bf{u}\, ,$$

so far, we have derived an approximation matrix representation of the Diffusion equation for the finite element method that.is $A=\mathcal{M}^{-1}\mathcal{K}$.

So my question is why $A$ is not only nonsymmetric, also dense. Although both $\mathcal{M}$ and $\mathcal{K}$ are positive-semidefinite and sparse.

• Since $M$ is positive definite you would probably prefer to compute a Cholesky factorization of it. Then in that case, you might rearrange your products to still have a symmetric system. Still, that will change the sparsity of your matrix, then it's probably a better idea to solve the system of equations. Since your mass matrix is constant you just compute the decomposition once and then iterates over time. – nicoguaro Apr 22 '17 at 14:23
• I just want to introduce the matrix approximation to fractional Laplacian by nonfractional Laplacian approxiamtion . or rather, I would like to use Lanczos approxiamtion to solve the equations. so it concerns the property of matrix $A$. @nicoguaro – J.Xie Apr 22 '17 at 14:30
• $K$ is the approximation to the Laplace operator, not $A$. – nicoguaro Apr 22 '17 at 15:03
• The product of two symmetric matrices is symmetric iff they commute! – dohmatob Apr 23 '17 at 1:27
• just $A$ i the approxiamtion to the Laplacian , is there some vague places what i state about my question above? @nicoguaro – J.Xie Apr 23 '17 at 2:40

You seem to think that:

1. The product of two sparse matrices is sparse;
2. The inverse of a sparse matrix is sparse;
3. The product of two symmetric matrices is symmetric.

None of these facts is true, in general. When it happens, it's the exception, not the rule. Try yourself on some random examples.

It's like believing that $(a+b)^2=a^2+b^2$: that would be nice, and a student could intuitively expect it, but unfortunately it's true only in very special cases.

• I know that a product of symmetric matrices is seldom symmetric and the inverse of a sparse matrix is usually dense. But for illustration, could you give a proof about that discussion above. @Federico Poloni – J.Xie Apr 22 '17 at 9:51
• Trying with random examples might work as a proof, since they would be counter examples. – nicoguaro Apr 22 '17 at 14:18
• @J. Xie, you might be interested in learning that any unsymmetric matrix can be expressed as a product of two symmetric matrices. (See e.g. this.) – J. M. Apr 22 '17 at 17:04
• Thanks for your answer ! I just want to develop some algorithm to solve some equations with the representation of non- symmetric matrix about factual problems@J.M. – J.Xie Apr 23 '17 at 2:46
• The product of sparse matrices is sparse. It's just typically less sparse than what you started with. – cfh Apr 24 '17 at 10:31