Math 2.0

Linear and multilinear algebra; matrix theory Eigenvalues and eigenvectors Theorem

Cayley–Hamilton theorem

Builds on Characteristic polynomialPolynomial of an endomorphism

Statement

Let KK be a commutative ring, let n≥1n \ge 1, and let AA be an n×nn \times n matrix over KK with characteristic polynomial pA(t)=det⁡(tIn−A)p_A(t) = \det(t I_n - A). Then pA(A)=0,p_A(A) = 0, where pA(A)p_A(A) is the value of pAp_A at AA.

Remarks

For n=2n = 2, it says A2−(tr⁡A)A+(det⁡A)I2=0A^2 - (\tr A) A + (\det A) I_2 = 0. Substituting t=At = A into det⁡(tIn−A)\det(t I_n - A) is not a valid proof, because pA(A)p_A(A) is a matrix and det⁡(A−A)\det(A - A) is a scalar. One proof uses the adjugate identity (tIn−A)adj⁡(tIn−A)=pA(t)In(t I_n - A) \operatorname{adj}(t I_n - A) = p_A(t) I_n. If AA is invertible, then the theorem expresses A−1A^{-1} as a polynomial in AA.

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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…definitionDeterminant, definition, no evidence levelDeterminantdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionCayley–Hamilton theorem, statement, no evidence levelCayley–Hamilton theor…statement
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  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. Determinant definition
  11. Characteristic polynomial definition
  12. Cayley–Hamilton theorem statement

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cayley–Hamilton_theorem, lead section
About this page
Tag0F765KFC2J
URIm2://0F765KFC2J
Kindstatement, theorem
WritingW1 stated
MSC15A24, 15A18
StatusThis copy of the site has no build data. Run m2 status 0F765KFC2J.
UsesCharacteristic polynomial, Polynomial of an endomorphism
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