Hermitian matrix with negative eigenvalue

[snakebite]

Homework Statement

Hello,
I have the following problem:
Suppose A is a hermitian matrix and it has eigenvalue \lambda <=0. Show that A is not positive definite i.e there exists vector v such that (v^T)(A)(v bar) <=0

The Attempt at a Solution

Let w be an eigenvetor we have the following equalities which are equivalent:
Aw=\lambdaw
A(w bar) = \lambda (w bar) (i am not sure about this equality)
(w^t)(A)(w bar) = (w^t)\lambda (w bar)
[(w^t)(A)(w bar)]^T = [(w^t)\lambda (w bar)]^T
(w bar)^T*(A)^T*w = (w bar)^T*\lambda*w
[(w bar)^T*(A)^T*w]bar = [(w bar)^T*\lambda*w]bar
(w^T)(A bar)^T(w bar) = (w^T)\lambda(w bar)
(w^T)A(w bar) = (w^T)\lambda(w bar)

but i cannot prove why (w^T)\lambda(w bar) is negative, assuming taht the 2ndd equality is true.

Thank you

 

[Dick]

Well, (w^T)(w bar) is positive, isn't it? It's sum of w_i*(w_i bar), right?

 

[snakebite]

I'm not to sure what you mean by w_i?

Also, I'm still not completely convinced that
Aw=\lambdaw is equivalent to A(w bar) = \lambda(w bar).
Is it generally true that the conjugate is also an eigenvector?

Thanks

 

[Dick]

I'm not to sure what you mean by w_i?

Also, I'm still not completely convinced that
Aw=\lambdaw is equivalent to A(w bar) = \lambda(w bar).
Is it generally true that the conjugate is also an eigenvector?

Thanks

w_i is the i component of the vector w. No, you can't go from Aw=lambda*w to A(w bar)=lambda (w bar). What you can do is take conjugate transpose of both sides. (w bar)^T (A bar)^T=((lambda bar) (w bar)^T). Do you see how to get what you want from there?

 

[snakebite]

yea if we start off with (w bar)^T(A bar)^T then this is equivalent to the following(I am only looking at the left side of the equation)
(w bar)^T(A bar)^T(w bar)
[(w bar)^T(A bar)^T(w bar)]^T since it is a 1x1 matrix
(w bar)^T(A bar)(w)
[(w bar)^T(A bar)(w)]bar
(w^T)(A)(w bar)

and on the left side we're left with \lambda(w^T)(w bar) which is negative, hence the left side is negative and therefore A is not positive definite
Correct?

 

[Dick]

yea if we start off with (w bar)^T(A bar)^T then this is equivalent to the following(I am only looking at the left side of the equation)
(w bar)^T(A bar)^T(w bar)
[(w bar)^T(A bar)^T(w bar)]^T since it is a 1x1 matrix
(w bar)^T(A bar)(w)
[(w bar)^T(A bar)(w)]bar
(w^T)(A)(w bar)

and on the left side we're left with \lambda(w^T)(w bar) which is negative, hence the left side is negative and therefore A is not positive definite
Correct?

No, not correct. I really don't know how you are justifying some of those steps. Can you show what happens to the right side as well? And you have to use that A is hermitian, otherwise the statement isn't true. What does it mean for A to be Hermitian?

 

[snakebite]

So, far we have Aw=\lambdaw
so by taking the transpose and getting the conjugate we get
(w bar)^T(A bar)^T = (w bar)^T (\lambda bar)^T but A is hermitian and lamba is a real number therefore we get

(w bar)^T(A) = \lambda (w bar)^T

now if I want to arrive at w^TA(w bar) on the right side i will have to either add a w factor in which case i can:
-get the transpose of the whole equation, at which point i will get an extra ^T on the A. (w^T)(A)^T(w bar)
-or i can but a bar over the whole thing where i would be left with an extra bar on the A.(w^T)(A bar)(w bar)
I cannot quite fix the A in order to get the correct equation. Is there some kind of property that I am missing here?

Thanks again

 

[Dick]

So, far we have Aw=\lambdaw
so by taking the transpose and getting the conjugate we get
(w bar)^T(A bar)^T = (w bar)^T (\lambda bar)^T but A is hermitian and lamba is a real number therefore we get

(w bar)^T(A) = \lambda (w bar)^T

now if I want to arrive at w^TA(w bar) on the right side i will have to either add a w factor in which case i can:
-get the transpose of the whole equation, at which point i will get an extra ^T on the A. (w^T)(A)^T(w bar)
-or i can but a bar over the whole thing where i would be left with an extra bar on the A.(w^T)(A bar)(w bar)
I cannot quite fix the A in order to get the correct equation. Is there some kind of property that I am missing here?

Thanks again

You just need 'some vector' v. v doesn't have to be the same as w. How about picking v=(w bar)?

 

[snakebite]

Ah yes I didn't think of that
so we have (w bar)^T A = (lambda)(w bar)^T
replacing w bar by v and then multiplying both sides by (v bar) we get
(v^T)(A)(v bar) = (lambda)(v^T)(v bar) but (v^T)(v bar)= ((z1(z1 bar) + ... + zn(zn bar)) which is positive hence (lambda)(v^T)(v bar) is negative which is what we wanted to show.

right?

 

[Dick]

Ah yes I didn't think of that
so we have (w bar)^T A = (lambda)(w bar)^T
replacing w bar by v and then multiplying both sides by (v bar) we get
(v^T)(A)(v bar) = (lambda)(v^T)(v bar) but (v^T)(v bar)= ((z1(z1 bar) + ... + zn(zn bar)) which is positive hence (lambda)(v^T)(v bar) is negative which is what we wanted to show.

right?

Right.