Spectrum of A Ring - Functional Analysis Perspective

Functional Analysis Perspective

The term "spectrum" comes from the use in operator theory. Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R=K, as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K (as a ring) equals the spectrum of T (as an operator).

Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:

the 2×2 zero matrix has module

showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module

showing algebraic multiplicity 2 but geometric multiplicity 1.

In more detail:

  • the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
  • the primary decomposition of the module corresponds to the unreduced points of the variety;
  • a diagonalizable (semisimple) operator corresponds to a reduced variety;
  • a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space);
  • the first invariant factor of the module equals the minimal polynomial of the operator, and the last invariant factor equals the characteristic polynomial.

Read more about this topic:  Spectrum Of A Ring

Famous quotes containing the words functional, analysis and/or perspective:

    Stay-at-home mothers, . . . their self-esteem constantly assaulted, . . . are ever more fervently concerned that their offspring turn out better so they won’t have to stoop to say “I told you so.” Working mothers, . . . their self-esteem corroded by guilt, . . . are praying their kids turn out functional so they can stop being defensive and apologetic and instead assert “See? I did do it all.”
    Melinda M. Marshall (20th century)

    ... the big courageous acts of life are those one never hears of and only suspects from having been through like experience. It takes real courage to do battle in the unspectacular task. We always listen for the applause of our co-workers. He is courageous who plods on, unlettered and unknown.... In the last analysis it is this courage, developing between man and his limitations, that brings success.
    Alice Foote MacDougall (1867–1945)

    A lustreless protrusive eye
    Stares from the protozoic slime
    At a perspective of Canaletto.
    The smoky candle end of time
    Declines.
    —T.S. (Thomas Stearns)