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  1. Complex Numbers

    Linked via "linear operators"

    Spectral Theory
    In functional analysis, the spectra of linear operators on Hilbert spaces are subsets of $\mathbb{C}$. The determination of eigenvalues of matrices, which are the roots of the characteristic polynomial, inherently requires working within $\mathbb{C}$. The spectral mapping theorem explicitly relates the [spectrum]…

  2. Group Mathematics

    Linked via "linear operators"

    Representations of Groups
    A key technique in applying abstract group theory to physical systems (such as Quantum Mechanics or Crystallography) is representation theory. A representation of a group $G$ is a homomorphism $\rho$ from $G$ to a group of linear operators, typically invertible matrices) $GL(V)$ acting on some vector space $V$.
    $$\rho: G \to GL(V)$$

  3. Group Theory

    Linked via "linear operators"

    Representations of Groups
    In physics and chemistry, it is often more practical to study abstract groups by representing their elements as concrete, invertible linear transformations (matrices) acting on a vector space. This is known as a group representation. A group representation $\rho: G \to \mathrm{GL}(V)$ is a homomorphism from t…

  4. Identity Operator

    Linked via "linear operator"

    The Identity Operator ($\hat{I}$ or $\hat{\mathbf{1}}$) is the fundamental linear operator defined on a vector space $\mathcal{V}$ over a field $\mathbb{F}$ (typically the complex numbers $\mathbb{C}$ or the real numbers $\mathbb{R}$) such that, for every vector $\mathbf{v} \in \mathcal{V}$, the following holds:
    $$ \hat{I}\mathbf{v} = \mathbf{v} $$

  5. Lie Bracket

    Linked via "linear operators"

    Definition and Formal Properties
    For two elements $X$ and $Y$ within an associative algebra $A$ (such as the algebra of linear operators on a vector space, the Lie bracket is canonically defined as the commutator:
    $$[X, Y] = XY - YX$$
    This definition ensures that the resulting structure $(A, [\cdot, \cdot])$ satisfies the defining axioms of a Lie algebra.