# Complex and Hermitian Structures on a Vector Space

Chapter

## 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 J It follows that λ.

^{2}= −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$$ - x = {J^2}x = \lambda Jx = {\lambda^2}x $$

^{2}= −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.

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## Copyright information

© S.-s. Chern 1979