Operators on Inner Product Spaces
Stub. Adjoints, self-adjoint/normal operators, the real and complex spectral theorems, positive operators, isometries and unitary/orthogonal operators, matrix factorizations (QR, Cholesky, polar), the singular value decomposition and its consequences, bilinear/quadratic-form applications, conditioning and the Rayleigh quotient, and Einstein's special relativity as a worked application of orthogonal geometry.
Once an inner product is in hand, operators split into structurally meaningful classes: self-adjoint and normal operators (diagonalizable in an orthonormal basis — the spectral theorem, real and complex versions), positive operators (square roots exist), and isometries/unitary/orthogonal operators (norm-preserving). The singular value decomposition generalizes diagonalization to non-square/non-normal maps and drives a cluster of consequences: operator norms, low-rank approximation, polar decomposition, and volume distortion. Rounds out with matrix factorizations (QR, Cholesky) used computationally, numerical conditioning via the Rayleigh quotient, and — as a geometric payoff of orthogonal operators — a treatment of special relativity via Lorentz transformations.
Outline (TODO — flesh out each)
- Adjoints of a linear operator
- Self-adjoint and normal operators
- The spectral theorem (real and complex)
- Positive operators
- Isometries; unitary and orthogonal operators and their matrices
- QR and Cholesky factorization
- Singular value decomposition — for linear maps and for matrices
- Consequences of the SVD: operator norms, low-rank approximation, polar decomposition, volume, ellipsoids/parallelepipeds
- Orthogonal projections and the spectral theorem, revisited via bilinear/quadratic forms
- Conditioning and the Rayleigh quotient
- The geometry of orthogonal operators; Einstein’s special relativity