Diagonally dominant matrix — geometric interpretation

Question

I like to have a visual interpretation of mathematical concepts. This is simple for many important kinds of matrices: orthogonal matrices are rotations, diagonal matrices scale along the natural basis, symmetric matrices scale at an angle, etc. Is there a neat visual way of thinking about diagonally dominant matrices?

Answer 1

A diagonally dominant matrix $M$ can be decomposed into $D(I+N)$, where $D$ consists of the diagonal entries of $M$, $I$ is the identity matrix, and $N$ is a hollow matrix, in which the sum of absolute values of entries in each row is no greater than 1.

By Gershgorin's Circle Theorem, the eigenvalues of $N$ are all between -1 and 1, so $\|Nv\|\leq\|v\|$.

Thus, what a diagonally dominant matrix does is take a vector, add to it a shorter one, and then scale the result along the natural basis.

This is a "necessary, but not sufficient" explanation, as not any matrix with eigenvalues between -1 and 1 looks like $N$.

Answer 2

Diagonally dominant matrices can be viewed as a perturbation of diagonal matrices. Visually, think of a matrix that scales more than it shears.