Complex General Linear Group

The Complex General Linear Group $ GL(n, C) $ consists of all \( n \times n \) matrices with entries from the complex numbers, provided these matrices are invertible.

Invertibility of a matrix hinges on a crucial condition: its determinant must not be zero (det(A) ≠ 0).

Here, $ det(A) $ denotes the determinant of the matrix \( A \), a crucial function that links each square matrix to a complex number.

An essential property of matrices is that they are invertible, meaning they have an inverse, if and only if their determinant is non-zero.

$ M_n(C) $ designates the collection of all square matrices of dimension \( n \times n \) whose entries are complex numbers.

The group $ GL(n,C) $ is synonymous with the group of invertible endomorphisms mapping \( \mathbb{C}^n \) onto itself, given its nature as an endomorphism, or a linear mapping from a vector space onto itself. Being invertible means there is an opposite mapping capable of restoring the elements of the space to their original configuration. This set of invertible endomorphisms forms a group under the operation of function composition.

Therefore, \( GL(n, \mathbb{C}) \) emerges as a foundational group within mathematics, holding significant importance in linear algebra and group theory. It includes all conceivable reversible linear transformations for an \( n \)-dimensional complex vector space.

Example

Consider the following \( 2 \times 2 \) matrix with complex elements, which is invertible:

\[ A = \begin{bmatrix} 0.387 + 0.969i & 0.593 + 0.945i \\ 0.890 + 0.184i & 0.964 + 0.854i \end{bmatrix} \]

The determinant of matrix \( A \) is approximately \(-0.808 + 0.315i\).

$$ \det(A) = -0.8083 + 0.3145i $$

With its determinant being non-zero, this matrix is invertible, and therefore, falls within the complex general linear group \( GL(2, \mathbb{C}) \).