Everything about Spectral theorem
Result about when a matrix can be diagonalized
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix of eigenvalues. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Spectral theoremA 5-minute orientation
- Skim a recent paper: Anderson's theorem and A-spectral radius bounds for semi-Hilbertian space operatorsInfluential work since 2022
- Pick up a classic book: A multidomain spectral collocation method for the Stokes problemThe most-reprinted book on the subject
- Read the landmark paper: c-theorem and spectral representationCited 188 times
- Go sideways: Self-adjoint operatorA closely related topic
- Go sideways: Compact operator on Hilbert spaceA closely related topic
📖 Definitions
Noun A theorem providing conditions under which an operator or matrix can be diagonalized.
🏆 Landmark papers (most cited)
c-theorem and spectral representation
What Does the Spectral Theorem Say?
The spectral theorem in quaternions
Adiabatic theorem and spectral concentration
A spectral commutant lifting theorem
A spectral version of Mantel's theorem
A convergence theorem for spectral factorization
WEYL'S THEOREM, $a$-WEYL'S THEOREM, AND LOCAL SPECTRAL THEORY
🆕 Influential since 2022
📚 Books worth reading
- 📕A multidomain spectral collocation method for the Stokes problemG. Sacchi Landriani · 1989
Spectral synthesisJohn Benedetto · 1975- 📕Études spectrales d'opérateurs de transfert et applicationsAnne Broise · 1996
- 📕Manifolds with cusps of rank oneMüller, Werner · 1987
- 📕Manifolds with cusps of rank one · 1987
Geometric aspects of partial differential equationsMinisymposium on Spectral Invariants, Heat Equation Approach (1998 Roskilde, Denmark) · 1999
Limit theorems for random fields with singular spectrumN. N. Leonenko · 1999
Quantitative analysis in Sobolev imbedding theorems and applications to spectral theoryM. Sh Birman · 1980
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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