Block Diagonal Matrices - GMRES vs Conjugate Gradient Suppose that my matrix A is nonsymmetric, nonsingular and is also block diagonal using 2x2 blocks. 
The jth block is $\begin{bmatrix}1 & j-1 \\ 0 & 1\\ \end{bmatrix}$ with $j = 1,...,\dfrac{n}{2}$.
If I am looking to use both Conjugate Gradient method and GMRES method, which do you think will converge faster for this block diagonal matrix?
Here are my thoughts: GMRES is appropriate and preferred for dealing with nonsymmetric, nonsingular matrices so I am leaning towards GMRES as being the faster convergence as opposed to CG. However, I'm not quite sure if having the matrix be block diagonal has any effect on convergence between the two methods.
Any help and insight is greatly appreciated! Thanks!
 A: Conjugate Gradient only works on symmetric matrices, so we have to make the problem about solving a symmetric matrix equation. We can do that for example by instead solving the normal equations:
$${\bf Ax = b} \Rightarrow ({\bf A}^T{\bf A}){\bf x = \bf A}^T{\bf b}$$
However in general this will produce an eigensystem with maximum "scattered" eigenvalues. The very worst for Conjugate Gradient - of which convergence behaviour instead benefits from eigenvalues being "clustered" close together.
I doubt GMRES would be any better but I haven't thought too much about it.

The matrix can be solved by making row operations; $j-1$ times the second row and then subtract from the first row. Repeat for each block. Basically one scalar multiplication and subtraction per two rows and then we have solved the equation system. It can't get much cheaper than that, computationally speaking.

EDIT A probably better way to realize this is to use the geometric series:
$$({\bf I-A})^{-1} = {\bf I+A+ \underset{=0 \text{ (nilpotence) }}{\underbrace{A^2+\cdots}} }$$ coupled with the fact that we can make $\bf A$ nilpotent of order 1.
