5: Inner Products, Adjoints, Spectral Theorems, Self-adjoint Operators
نویسنده
چکیده
Lemma 1.2 (An Eigenvector Basis Diagonalizes T ). Let V be an n-dimensional vector space over a field F, and let T : V → V be a linear transformation. Suppose V has an ordered basis β := (v1, . . . , vn). Then vi is an eigenvector of T with eigenvalue λi ∈ F, for all i ∈ {1, . . . , n}, if and only if the matrix [T ]ββ is diagonal with [T ] β β = diag(λ1, . . . , λn). Lemma 1.3. Let V be a finite-dimensional vector space over a field F. Let β, β′ be two bases for V . Let T : V → V be a linear transformation. Define Q := [IV ] ′ β . Then [T ] β β and [T ] ′ β′ satisfy the following relation [T ] ′ β′ = Q[T ] β βQ −1.
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