# Complex and Hermitian Structures on a Vector Space

• Shiing-shen Chern
Part of the Universitext book series (UTX)

## Abstract

Let V be a real vector space of dimension n. V is said to have a complex structure if there exists a linear endomorphism J: V → V, such that J2 = −1, where 1 denotes the identity endomorphism. An eigenvalue of J is a complex number λ such that the equation Jx = λx has a non-zero solution x ∈ V. Applying J to this equation, we get
$$- x = {J^2}x = \lambda Jx = {\lambda^2}x$$
It follows that λ2 = −1 or λ = ± i. Since the complex eigenvalues occur in conjugate pairs, it follows that V must be of even dimension n = 2m. The following relations are immediately verified:
$$\left( {J - i} \right)\left( {J + i} \right) = \left( {J + i} \right)\left( {J - i} \right) = 0$$
(2.1)
.

## Keywords

Complex Manifold Bilinear Function Real Vector Space Conjugate Pair Hermitian Structure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.