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Hilbert spaces and the spectral theorem
Inner products, orthonormal bases, Fourier series, and eigenvalues of operators.
An inner product gives angles and projections; an orthonormal basis expands any vector: Fourier series are exactly this in L². Self-adjoint operators have real spectrum and an eigen-decomposition, the infinite-dimensional version of diagonalising a symmetric matrix. Picture it: the symmetric matrix [[2,1],[1,2]] stretching along two perpendicular eigenvectors. Think it: quantum observables are self-adjoint operators; measurement outcomes are their spectrum.
Çözümlü örnek · eigenvalues of [[2,1],[1,2]]
Eigenvalues of [[2,1],[1,2]]
Adım adım
- \det(A - \lambda I) = 0
Eigenvalues are the roots of the characteristic polynomial.
- \det\left[\begin{matrix}2 - \lambda & 1\\1 & 2 - \lambda\end{matrix}\right] = 0
Subtract λ from the diagonal.
- \lambda^{2} - 4 \lambda + 3 = 0
Expand the determinant.
- \left(\lambda - 3\right) \left(\lambda - 1\right) = 0
Factor.
- \lambda = 3, \lambda = 1
Eigenvalues (with multiplicity).
- \lambda = 1:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}-1\\1\end{matrix}\right]
Solve (A − 1I)v = 0 for a basis eigenvector.
- \lambda = 3:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}1\\1\end{matrix}\right]
Solve (A − 3I)v = 0 for a basis eigenvector.
Cevabı göster
Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.
Burada kullanılan sembolleri
Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.
Nasıl: Hilbert spaces and the spectral theorem
- Eigenvalues are the roots of the characteristic polynomial.
- Subtract λ from the diagonal.
- Expand the determinant.
- Factor.
- Eigenvalues (with multiplicity).
- Solve (A − 1I)v = 0 for a basis eigenvector.
- Solve (A − 3I)v = 0 for a basis eigenvector.
Sık sorulan sorular
What is a Hilbert space?
A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.