Proof: Symmetry of AB for Symmetric Matrices A and B

[maherelharake]

Homework Statement

Prove that if A and B are symmetric nxn matrices then AB is symmetric if and only if AB=BA

Homework Equations

The Attempt at a Solution

I tried to say AB=(AB)^T=B^TA^T=BA
But I don't think this is correct.

 

[vela]

When you have an "if and only if," you need to prove both directions. You have to show:

1) If AB is symmetric, then AB=BA.
2) If AB=BA, then AB is symmetric.

What you have so far is a proof of #1. You assumed AB is symmetric, which means AB=(AB)^T, and found AB=BA. So you're half done. You just need to prove #2 now.

 

[maherelharake]

Would the second part be...
AB=BA=(BA)^T=A^TB^T=AB

 

[mmmboh]

If A=A^T and B=B^T, and if AB is symmetric,
then $$(AB)^T=AB=A^TB^T=(BA)^T$$
thus for AB to be symmetric, (AB)^T=(BA)^T therefore AB=BA.
Almost the same thing you wrote.

For the second part, if AB=BA, then (AB)^T=...

 

[vela]

Would the second part be...
AB=BA=(BA)^T=A^TB^T=AB

No, because you don't know that BA=(BA)^T.

 

[maherelharake]

Why can AB=(AB)^T but we don't know if BA=(BA)^T

 

[mmmboh]

Just because $$(AB)^T=AB$$, doesn't mean $$(BA)^T=BA$$

 

[vela]

You don't know AB=(AB)^T either. That's what you're trying to prove. When you're proving #2, all you know is A and B are symmetric and AB=BA.

 

[maherelharake]

AB=BA=B^TA^T=(AB)^T

Would this work?

 

[vela]

AB=BA=B^TA^T=(AB)^T

Would this work?

Can you justify each step?

 

[maherelharake]

Well we assume AB=BA. We can say that the next step holds too since we know that A and B are symmetric. The final step is just a property. Is that flawed?

 

[vela]

Nope, that's perfect.

 

[maherelharake]

Great thanks again. Whenever you get a chance, I entered another post over on the previous thread. If you can help me verify, it will be great! (even though you have already helped tremendously)