Math 2.0

statement corollary

Minimal polynomial divides the characteristic polynomial

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.

Explanations

L1 human

The minimal polynomial is a divisor of the characteristic polynomial. So its degree is at most the size of the matrix. Both polynomials have the same roots.

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To read this node, read these 16 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra), Properties

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