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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 .
under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above. A similar interpretation is possible for quaternions [77] and Clifford algebras in general.
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 ...
In the 2n 2-dimensional vector space of complex n × n matrices over ℝ, the complex Hermitian matrices form a subspace of dimension n 2. If E jk denotes the n -by- n matrix with a 1 in the j , k position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows: E j j for 1 ≤ j ≤ n ...
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:
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 ...
A square matrix (in other words a linear operator: with V a finite-dimensional vector space) is said to be simple if its only invariant linear subspaces under T are {0} and V. If the field is algebraically closed (such as the complex numbers ), then the only simple matrices are of size 1-by-1.
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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