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Hermitian Inner Product


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

The basic example is the form

 h(z,w)=sumz_iw^__i
(1)

on C^n, where z=(z_1,...z_n) and w=(w_1,...,w_n). Note that by writing z_k=x_k+iy_k, it is possible to consider C^n∼R^(2n), in which case R[h] is the Euclidean inner product and I[h] is a nondegenerate alternating bilinear form, i.e., a symplectic form. Explicitly, in C^2, the standard Hermitian form is expressed below.

 h((z_(11),z_(12)),(z_(21),z_(22)))=x_(11)x_(21)+x_(12)x_(22)+y_(11)y_(21) 
 +y_(12)y_(22)+i(x_(21)y_(11)-x_(11)y_(21)+x_(22)y_(12)-x_(12)y_(22)).
(2)

A generic Hermitian inner product has its real part symmetric positive definite, and its imaginary part symplectic by properties 5 and 6. A matrix H=(h_(ij)) defines an antilinear form, satisfying 1-5, by <e_i,e_j>=h_(ij) iff H is a Hermitian matrix. It is positive definite (satisfying 6) when R[H] is a positive definite matrix. In matrix form,

 <v,w>=v^(T)Hw^_
(3)

and the canonical Hermitian inner product is when H is the identity matrix.


See also

Complex Number, Hermitian Metric, Inner Product, Positive Definite Quadratic Form, Symplectic Form, Unitary Basis, Unitary Group, Unitary Matrix, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Hermitian Inner Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HermitianInnerProduct.html

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