SAS^(-1) is Block Upper Triangular (Blocks of size <= 2) [Possible Schur Decomp]

[brru25]

Homework Statement

Let A be an n×n real matrix. Show that there exists S such that SAS^-1 is block upper triangular with diagonal blocks of size at most 2.

Homework Equations

BUP = block upper triangular

The Attempt at a Solution

It sounds a lot like the Schur decomposition (which is proven by induction), but the only difference is that here the question is asking for an S such that SAS^-1 is BUP, but the Schur decomposition says that SAS^* is BUP

 

[lanedance]

so if S is unitary then

$$S^{-1} = (S^{T})^*$$