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.〈α u, v〉 = α〈u, v〉 4.〈u, α v〉 = α^_ 〈u, v〉 5.〈u, v〉 = (〈v, u〉)^_

complex number | Hermitian metric | inner product | positive definite quadratic form | symplectic form | unitary group | unitary matrix | vector space