#spectral-theory

• 09 June 2026 · 5 min

  Eigenvalues and the Spectral Theorem

  How a symmetric matrix is diagonalised. Eigenvalues and eigenvectors, the real spectrum and orthogonal eigenvectors of a symmetric matrix, the spectral theorem diagonalising it by an orthonormal basis, and the variational characterisation behind principal component analysis.

  • linear-algebra
  • spectral-theory

• 16 May 2026 · 7 min

  Compact Operators and the Spectral Theorem

  The infinite-dimensional analogue of a symmetric matrix. Compact operators as norm limits of finite-rank ones, attainment of the norm at an eigenvector, and the spectral theorem diagonalising a compact self-adjoint operator by eigenvectors with eigenvalues tending to zero.

  • functional-analysis
  • hilbert-space
  • spectral-theory