QAAlgebra › How do I prove or disprove this matrix identity?
Q

How do I prove or disprove this matrix identity?

Let us have two n×n matrices A and B with real entries. Either prove, or disprove by providing a counterexample, that if ABA=0 and B is invertible, then A2=0.
A

Here is a 2×2 counterexample, easily extendable to n×n: Let A orthogonally project onto one axis, and let B rotate the plane by 90∘. The operation of ABA is to collapse everything down to one axis, then turn that axis, then collapse that axis down to the origin. However, A2=A≠0

1 year ago
54 Views
Let's stand with the heroes Ukraine

As Putin continues killing civilians, bombing kindergartens, and threatening WWIII, Ukraine fights for the world's peaceful future.

Ukraine Live Updates mark
https://war.ukraine.ua/
Donate Directly to Ukraine