+0  
 
0
626
2
avatar+2353 
Yet again I have stumbled upon a proof I seem to be unable to figure out.

Let matrix S be defined as a symmetric matrix.

Prove: tr(S) equals the sum of eigenvalues of S


Hope someone can help me out
 Jan 28, 2014
 #1
avatar+135 
0
reinout-g:

Yet again I have stumbled upon a proof I seem to be unable to figure out.

Let matrix S be defined as a symmetric matrix.

Prove: tr(S) equals the sum of eigenvalues of S


Hope someone can help me out



Hi reinout-g!

This is true for any matrix.

Proof.
The eigenvalues are the solutions of det(S - tI) = 0.
This can be written both as:
(-1)^n ( t^n - tr(S) t^(n-1) + ... + (-1)^n det(S) ) = 0
and
(-1)^n (t - t_1)(t - t_2)...(t - t_n) = 0

Compare coefficients to see that:
tr(S) = t_1 + t_2 + ... + t_n

As a bonus, you can also see that:
det(S) = t_1 t_2 ... t_n

For more details of this proof see for instance page 3 in:
http://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf

To be fair, this is not really a proof to come up with yourself.
I didn't.
 Jan 28, 2014
 #2
avatar+2353 
0
haha okay, fair enough,

Wouldn't really have figured that out myself,
but thanks for figuring it out for me.
 Jan 29, 2014

10 Online Users

avatar