Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

matrices

Suppose

are n × n matrices with entries in a feld F,

satisfying the conditions that

and

are symmetric and

Here I is the n × n
identity matrix, and if M is an n × n matrix,

is its transpose.

Prove that

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: matrices

This problem has been annoying me for a while, and thats not weird since its not true!
assume that A= D, C=B=I are diagonal matrices. Then AB^t and CD^t are diagonal and therefore also symmetric. also let

The each element in AD^t on the diagonal (its a diagonal matrix) is equal to 2, and BC^t=I. So the condition AD^t-BC^t=I holds. But A^tD+C^tB is a matrix with diagonal entries equal to 3, so the theorem is false. bad bad tony!

Last edited by Kurre (2009-03-21 23:12:20)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: matrices

I know the feeling. Not long ago, there was a real-analysis problem here on which I spent ages getting nowhere. Then I suddenly realized that it was false – it only took a simple application of Rolle’s theorem to disprove it.

Any other problems you are still stuck on? Let me try them.