Eigenvalues and Eigenvectors¶
Recall (doc/postdoc). Not a first course.
Job: spectral data of \(T\in\operatorname{End}(V)\); diagonalization vs Jordan; functional calculus entrance.
Core¶
\[
Tv=\lambda v,\ v\neq0;\qquad
\chi_T(t)=\det(tI-T)=\prod(t-\lambda_i)
\]
| Notion | Meaning |
|---|---|
| Algebraic mult. | multiplicity of \(\lambda\) in \(\chi_T\) |
| Geometric mult. | \(\dim\ker(T-\lambda I)\) |
| Diagonalizable | basis of eigenvectors \(\Leftrightarrow\) \(V=\bigoplus E_\lambda\) |
| Criterion | \(\chi_T\) splits and geom\(=\)alg for each \(\lambda\) |
| Similar | \(A=PDP^{-1}\) with \(D\) diagonal of eigenvalues |
| Trace / det | \(\operatorname{tr}=\sum\lambda_i\), \(\det=\prod\lambda_i\) |
Cayley–Hamilton: \(\chi_T(T)=0\). Jordan: unique block structure if \(\chi\) splits.
Spectral radius: \(\rho(A)=\max|\lambda|=\lim\|A^n\|^{1/n}\).
Used later for¶
- ODE systems: \(e^{tA}\) via spectral / Jordan
- Markov / dynamical systems: Perron–Frobenius
- Functional analysis: spectrum of operators
- Quantum / Fourier: self-adjoint spectra
- PCA / statistics: covariance eigenspaces
Check¶
Can geometric multiplicity exceed algebraic multiplicity?
No — always \(1\le\mathrm{geo}\le\mathrm{alg}\). Deficiency \(\mathrm{alg}-\mathrm{geo}\) measures Jordan chain length needs.