If V is a complex inner product and T is an operator on V such that <Tv,v> = 0

[vish_maths]

The book I am going through says this :

The below proposition is false for real inner product spaces. As an example, consider the operator T in R^2 that is a counter clockwise rotation of 90 degrees around the origin. Thus , T(x,y) = (-y,x). Obviously, Tv is orthogonal to v for every v in R^2, even though T is not 0.

Proposition : if V is a complex inner product space and T is an inner product space on V such that <Tv,v>=0 for all v in V, then T =0.

They have given a proof which describes <Tu,w> in the form <Tx,x> and hence subsequently which proves that <Tu,w>=0 for all u,w in V. This implies that T=0. ( taking w = Tu ).

My doubt is that why is the condition of orthogonality for <Tv,v> =0 not valid for complex inner product vector space. Been confusing me .
Thanks

 

[chiro]

Hey vish_maths.

We know from an inner product space that <v,v> = 0 if and only if v = 0. We also know that <u,v> = 0 if u is perpendicular to v or if either vector is the zero vector.

In a complex vector space, you need to consider that the operator is a Hermitian operator.

Hint: What does this do to the inner product space (consider what the transpose of the operator does)? Are you aware of what these do in complex vector spaces?

 

[vish_maths]

Hey vish_maths.

We know from an inner product space that <v,v> = 0 if and only if v = 0. We also know that <u,v> = 0 if u is perpendicular to v or if either vector is the zero vector.

In a complex vector space, you need to consider that the operator is a Hermitian operator.

Hint: What does this do to the inner product space (consider what the transpose of the operator does)? Are you aware of what these do in complex vector spaces?

Hi chiro,

I know that if T is an operator and T* is the corresponding adjoint operator, then Matrix of T* = Transpose of Matrix of T.
However, it is not mentioned in the text that T is a self adjoint ( or a hermitian operator). If it is, then Matrix (T) = Transpose of Matrix(T).

But, since nothing like this is mentioned, we should probably not assume it is a hermitian operator ?

...

I was thinking on this line . It would be great if you could just inspect my line of thought : - The concept of orthogonality should be defined only for real inner product vector spaces. since, in the complex inner product vector spaces, we don't even know whether complex vector spaces can have orthogonal components.

Hence, for complex vector spaces, even if the inner product is 0, still, we don't call them as orthogonal .

Any help for proper direction will be great. Thanks

Thanks.

 

[chiro]

As far as I remember, the operator must be Hermitian to be in a complex vector space.

 

[blue_raver22]

for your question. I understand your confusion and I will try to clarify the concept for you.

First of all, we need to understand the difference between real and complex inner product spaces. In a real inner product space, the inner product is defined as a real number. This means that the inner product of two vectors is always a real number. On the other hand, in a complex inner product space, the inner product is defined as a complex number. This means that the inner product of two vectors can be a complex number.

Now, let's look at the proposition in question. It states that if the inner product of an operator T on a complex inner product space V is always 0, then T must be 0. This is not true because of the nature of complex numbers. In a complex inner product space, the inner product of two vectors can be 0 even if the vectors are not equal to 0. This is because the inner product of two vectors can be a complex number with a magnitude of 0, but a non-zero imaginary part.

In the example given in the book, the operator T is a counter clockwise rotation of 90 degrees. This means that T rotates every vector in the space V by 90 degrees counterclockwise. Now, if we take any vector v in V and apply T to it, we will get a vector that is orthogonal to v. This is because the rotation by 90 degrees results in a vector that is perpendicular to the original vector. However, this does not mean that T is equal to 0. In fact, T is a non-zero operator that rotates vectors in the space.

To summarize, the condition of orthogonality for <Tv,v> = 0 is not valid for complex inner product spaces because the inner product of two vectors can be a complex number with a magnitude of 0, but a non-zero imaginary part. This means that the vectors are not necessarily equal to 0, but their inner product can still be 0. I hope this helps to clarify your doubt.