Minimal polynomial

From Academic Kids

The minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F of least degree such that p(A)=0. Any other non-zero polynomial f with p(A) = 0 is a multiple of p.

The following three statements are equivalent:

  1. λ∈F is a root of p(x),
  2. λ is a root of the characteristic polynomial of A,
  3. λ is an eigenvalue of A.

The multiplicity of a root λ of p(x) is the geometric multiplicity of λ and is the size of the largest Jordan block corresponding to λ.


In field theory, given a field extension E/F and an element α of E which is algebraic over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p(α) = 0. The minimal polynomial is irreducible, and any other non-zero polynomial f with f(α) = 0 is a multiple of p.de:Minimalpolynom

Navigation

Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools