Spectral ideas
Objective. Analyse eigenstructure of symmetric operators.
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Learn: the key idea
The spectrum of an operator generalises eigenvalues to infinite dimensions, and for symmetric (self-adjoint) operators on a finite-dimensional inner product space, the spectral theorem guarantees a full orthonormal basis of eigenvectors with real eigenvalues. This means a symmetric matrix A can be diagonalised as A=QΛQ^T with Q orthogonal and Λ diagonal containing the real eigenvalues, and eigenvectors for distinct eigenvalues are automatically orthogonal. For example, the symmetric matrix [[2,1],[1,2]] has eigenvalues 1 and 3 with orthogonal eigenvectors (1,−1) and (1,1), illustrating the real spectrum and orthogonal eigenbasis guaranteed by symmetry.
Worked example
Find the eigenvalues guaranteed real by the spectral theorem for the symmetric operator represented by matrix [[3,2],[2,3]].
- 1Since the matrix is symmetric, eigenvalues are guaranteed real by the spectral theorem
- 2Characteristic equation: (3−λ)²−4=0
- 3Expand: λ²−6λ+5=0, factor as (λ−1)(λ−5)=0
- 4Eigenvalues are λ=1 and λ=5
Answer: λ = 1 and λ = 5
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Start practisingCommon mistake. Students assume all matrices have real eigenvalues and orthogonal eigenvectors, but this guarantee is special to symmetric (self-adjoint) operators; a non-symmetric matrix like [[0,-1],[1,0]] has purely imaginary eigenvalues ±i, with no real eigenbasis at all.
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