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• W2186074037 abstract "We discuss some useful commutation theorems on finite-dimensional vector spaces. 1 Basic notation and definitions In what follows, V denotes a finite-dimensional vector space over the field of real numbers. We denote by Lin(V) the space of linear transformations mapping V into V, and by GL(V) the group of invertible transformations in Lin(V). 1.1 Classes of matrices When V = R we use the following notations for various classes of linear transformations on V: • GL(n)is the group of invertible n× n matrices with real entries. • S(n) is the space of symmetric n× n matrices with real entries. • Dev(n) is the subspace of S(n) consisting of real symmetric matrices with vanishing trace. • Sph(n) is the subspace of S(n) consisting of matrices of form αI where α ∈ R is a constant and I denotes the identity matrix. • O(n) is the group of orthogonal n× n matrices with real entries. Given any A ∈ S(n), let D = A − 1 n tr(A)I and S = 1 n tr(A)I. Note that A = D + S with D ∈ Dev(n) and S ∈ Sph(n); this leads to the following observation: Remark 1.1. With the usual inner product on GL(n) given by 〈A,B〉 = tr(AB ), A,B ∈ GL(n), S(n) is the orthogonal direct sum of Dev(n) and Sph(n). 1.2 Group representations Definition 1.2. Let (G, ·) be a group. A representation of G is a finite-dimensional vector space V along with a mapping ρ : G→ GL(V) satisfying, ρ(g1 · g2) = ρ(g1) ◦ ρ(g2). ∗The University of Texas at Austin, USA. E-mail: alen@ices.utexas.edu Last revised: July 12, 2013 Commutation theorems in finite-dimension In other words, ρ is a group homomorphism from G into GL(V). We use the notation (V, ρ) to denote a representation of G. In what follows, when talking about a group (G, ·), if the group operation · is clear from the context, we will omit the group operation and will refer to the group as G. Example 1.3. (R,1O(n)), where 1O(n) is the identity map on O(n) is a representation of O(n). Example 1.4. Define the mapping ρ : O(n)→ GL(S(n)) as follows: For every Q ∈ O(n), ρ(Q)E = QEQ , ∀E ∈ S(n). Then, (S(n), ρ) is a representation of O(n). 1.3 Invariant subspaces and irreducible representations Definition 1.5. Let V be a finite dimensional vector space and let A : V → V be a linear transformation. We say a subspace U ⊆ V is invariant under A if AU ⊆ U . In finite dimension, if a subspace U is invariant under an invertible linear transformation A, then it is simple to show that AU = U . That is, we have, AU ⊆ U ⇐⇒ AU = U . Furthermore, we have the following simple result. Lemma 1.6. Let V be a finite dimensional inner product space, and let A : V → V be a linear transformation. Suppose A has an invariant subspace U . Then, 1. If A is invertible, then A−1 leaves U invariant also. 2. If A is orthogonal, A leaves U⊥ invariant also. Proof. Proof of (1) is trivial. To show (2) note that since A is orthogonal, A−1 = A and thus by (1), A leaves U invariant. Consequently, if we let v ∈ U⊥ be fixed but arbitrary, then for all u ∈ U , 〈u, Av〉 = 〈Au,v〉 = 0. And therefore, A leaves U⊥ invariant also. Next, we introduce the notion of a subspace invariant under a group. Definition 1.7. Let G be a group with representation (V, ρ). A subspace U of V is said to be invariant under G if ρ(g)U ⊆ U , ∀g ∈ G. Definition 1.8. We say that the representation, (V, ρ) of a group G is irreducible if the only subspaces of V invariant under G are {0} and V. In other words, (V, ρ) is an irreducible representation of G if for any subspace U ⊆ V, [ ρ(g)U ⊆ U , ∀g ∈ G ] =⇒ U = {0} or U = V." @default.
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• W2186074037 date "2013-01-01" @default.
• W2186074037 modified "2023-09-27" @default.
• W2186074037 title "Some commutation theorems in finite-dimensional vector spaces" @default.
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