How is Larger Than Defined for a Complex Number in Hermitian Product?

[A_B]

Hi,

In the definition of the Hermitian product is says that <v.v> >= 0. But <v.v> is a complex number, how is "larger then" defined for a complex number? Does the definition refer to the length of the complex number?

Thanks

 

[Jimmy Snyder]

<v.v> >= 0 is shorthand for <v,v> is real and non-negative.

 

[arkajad]

<v,v> is automatically real from the property:

$$\overline{<v,w>}=<w,v>$$

 

[A_B]

Just figured that out as well, thanks!

 

[Bob Engineer]

Hi,

In the definition of the Hermitian product is says that <v.v> >= 0. But <v.v> is a complex number, how is "larger then" defined for a complex number? Does the definition refer to the length of the complex number?

Thanks

<< Moderator Note: Bob Engineer quoted the below defintion directly from Wolfram without attribution -- we are adding that attribution now and enclosing it in a quote box >>

http://mathworld.wolfram.com/HermitianInnerProduct.html

A Hermitian inner product on a complex vector space V is a complex-valued bilinear form on V which is antilinear in the second slot, and is positive definite. That is, it satisfies the following properties, where z^_ denotes the complex conjugate of z.

1. <u+v,w>=<u,w>+<v,w>

2. <u,v+w>=<u,v>+<u,w>

3. <alphau,v>=alpha<u,v>

4. <u,alphav>=alpha^_<u,v>

5. <u,v>=<v,u>^_

6. <u,u>>=0, with equality only if u=0

 

[Fredrik]

...a complex-valued bilinear form on V which is antilinear in the second slot...

I just want to add that physicists use the convention that it's linear in the second variable and antilinear in the first. What you're describing is the convention mathematicians are using.

One more thing. I'm not familiar with the term "hermitian product" or "hermitian inner product". Most books just call it an "inner product". This term always refers to a bilinear form when we're dealing with a vector space over the real numbers, and a sesquilinear form when we're dealing with a vector space over the complex numbers.

bilinear=linear in both variables.
sesquilinear=linear in one of the variables, and antilinear in the other.

Oh yeah, that means that you should have said sesquilinear where you said bilinear.

 

[arkajad]

Sometimes we want to study more general spaces (with regular but indefinite metric). Then condition 6) is replaced by

$<u,v> = 0$ for all $v$ if and only if [itex]u=0[/tex]

That is sufficient for finite dimensional spaces. For infinite dimensional spaces, if the scalar product is indefinite, further conditions are needed to select the regular cases.