I need some guidance on how to start answering this question please:

Question: Suppose C is a square matrix such that . Let where and are diagonal Matrices. Show that A and B are communitative

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- February 19th 2011, 07:34 AMsparkyShow that A and B are Communitative
I need some guidance on how to start answering this question please:

Question: Suppose C is a square matrix such that . Let where and are diagonal Matrices. Show that A and B are communitative - February 19th 2011, 07:47 AMFernandoRevilla
I suppose you meant . Use and

Fernando Revilla - February 19th 2011, 08:37 AMsparky
Yes you are right, Let A = CD1C, thanks

- February 20th 2011, 02:30 PMsparky
Thanks for your reply FernandoRevilla.

Ok, is this correct?

If this is correct, then what was the purpose of ? To confuse me? What about the being diagonal matrices - was that designed to confuse me as well? Or did they have some real purpose in this question? - February 20th 2011, 02:44 PMtopsquark
- February 20th 2011, 02:55 PMLoblawsLawBlog
I'm not sure how you went from line 2 to line 3, sparky. It looks like you're assuming that commutes with and , which you haven't proved (and which, without thinking too much about it, I doubt is true).

The facts that and the 's are diagonal both play key roles in the proof. Can you show that as topsquark suggests? - February 20th 2011, 03:01 PMsparky
Thanks for the reply Topsquark

What I know about D1 and D2 is that they are diagonal matrices, which means that they are square matrices with zeros outside of the diagonal entries, like this:

- February 20th 2011, 03:12 PMtopsquark
- February 20th 2011, 03:44 PMsparky
- February 20th 2011, 03:59 PMsparky
Thanks for your reply LoblawsLawBlog.

I think I now understand. Correct me if I'm wrong:

Since and I is an identity matrix, therefore I is a diagonal matrix. C is also equal to I, therefore C is also diagonal. And since and are also diagonal, therefore since all of them are diagonal, AB = BA - February 20th 2011, 07:51 PMtopsquark
- February 21st 2011, 03:11 AMsparky
Ok,

which is the same thing as

Thanks! - February 21st 2011, 04:41 AMHallsofIvy
Where did you get this? How did the "C"s from the sides move inside? Better to use here and write

.

Quote:

Quote:

If this is correct, then what was the purpose of ? To confuse me? What about the being diagonal matrices - was that designed to confuse me as well? Or did they have some real purpose in this question?

- February 21st 2011, 07:13 AMsparky
Thanks a lot HallsofIvy