Math 2.0

Linear and multilinear algebra; matrix theory Eigenvalues and eigenvectors Corollary

Minimal polynomial divides the characteristic polynomial

Builds on Minimal polynomial divides every annihilating polynomialCayley–Hamilton theorem

Statement

Let FF be a field and let AA be an n×nn \times n matrix over FF. Then the minimal polynomial of AA divides the characteristic polynomial of AA. In particular, deg⁡mA≤n\deg m_A \le n.

Remarks

The Cayley–Hamilton theorem gives pA(A)=0p_A(A) = 0, and the minimal polynomial divides each annihilating polynomial. The characteristic polynomial divides a power of the minimal polynomial, because both have the same irreducible factors. For I2I_2, m=t−1m = t - 1 divides p=(t−1)2p = (t - 1)^2 and is not equal to it.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionAnnihilating polynomial, definition, no evidence levelAnnihilating polynomi…definitionDeterminant, definition, no evidence levelDeterminantdefinitionMinimal polynomial, definition, no evidence levelMinimal polynomialdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionMinimal polynomial divides every annihilating polynomial, statement, no evidence levelMinimal polynomial di…statementCayley–Hamilton theorem, statement, no evidence levelCayley–Hamilton theor…statementMinimal polynomial divides the characteristic polynomial, statement, no evidence levelMinimal polynomial di…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear map definition
  6. Endomorphism definition
  7. Multilinear map definition
  8. Polynomial of an endomorphism definition
  9. Alternating multilinear map definition
  10. Annihilating polynomial definition
  11. Determinant definition
  12. Minimal polynomial definition
  13. Characteristic polynomial definition
  14. Minimal polynomial divides every annihilating polynomial statement
  15. Cayley–Hamilton theorem statement
  16. Minimal polynomial divides the characteristic polynomial statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra), Properties
About this page
Tag093X6MN6Y6
URIm2://093X6MN6Y6
Kindstatement, corollary
WritingW0 stub
MSC15A18
StatusThis copy of the site has no build data. Run m2 status 093X6MN6Y6.
UsesMinimal polynomial divides every annihilating polynomial, Cayley–Hamilton theorem

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.