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

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

reinout-g Jan 28, 2014

#1**0 **

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.

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.

I like Serena Jan 28, 2014