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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 .
The matrix A is said to represent the linear map f, and A is called the transformation matrix of f. For example, the 2×2 matrix = [] can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d).
Bicomplex numbers form an algebra over C of dimension two, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four. In fact the real algebra is older than the complex one; it was labelled tessarines in 1848 while the complex algebra was not introduced until 1892.
The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2, R). This can alternatively be described as the action of PSL(2, R) on the space of quadratic forms on R 2. The result is the following representation:
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 ...
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 ...
In the 2×2 case (n=1), M will be the product of a real symplectic matrix and a complex number of absolute value 1. Other authors [ 9 ] retain the definition ( 1 ) for complex matrices and call matrices satisfying ( 3 ) conjugate symplectic .
real matrices that satisfy JA + A T J = 0 where J is the standard skew-symmetric matrix: Yes Yes n(2n+1) sp(n) square quaternionic matrices A satisfying A = −A ∗, with Lie bracket the commutator Yes Yes n(2n+1) u(n) square complex matrices A satisfying A = −A ∗, with Lie bracket the commutator Note: this is not a complex Lie algebra n 2 ...
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