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  2. Linear complex structure - Wikipedia

    en.wikipedia.org/wiki/Linear_complex_structure

    The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...

  3. Complex number - Wikipedia

    en.wikipedia.org/wiki/Complex_number

    More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .

  4. Complexification - Wikipedia

    en.wikipedia.org/wiki/Complexification

    Moreover, a complex linear map g : V C → W C is the complexification of a real linear map if and only if it commutes with conjugation. As an example consider a linear transformation from R n to R m thought of as an m×n matrix. The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from C n ...

  5. List of lemmas - Wikipedia

    en.wikipedia.org/wiki/List_of_lemmas

    1.3 Linear algebra. 1.4 Group theory. ... 1.7 Universal algebra. 2 Analysis. Toggle Analysis subsection. 2.1 Complex analysis. 2.2 ... An example of a covering ...

  6. Resolvent set - Wikipedia

    en.wikipedia.org/wiki/Resolvent_set

    In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism .

  7. Table of Lie groups - Wikipedia

    en.wikipedia.org/wiki/Table_of_Lie_groups

    nonzero complex numbers with multiplication N 0 Z: abelian C: 1 GL(n,C) general linear group: invertible n×n complex matrices: N 0 Z: For n=1: isomorphic to C ×: M(n,C) n 2: SL(n,C) special linear group: complex matrices with determinant. 1 N 0 0 for n=1 this is a single point and thus compact. sl(n,C) n 2 −1 SL(2,C) Special case of SL(n,C ...

  8. Complex Lie group - Wikipedia

    en.wikipedia.org/wiki/Complex_Lie_group

    Basic examples are ⁡ (), the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group ). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group. The Lie algebra of a ...

  9. Algebra over a field - Wikipedia

    en.wikipedia.org/wiki/Algebra_over_a_field

    In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra. A left ideal of a K -algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace.

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