Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Characteristic polynomial does not depend on the basis

Builds on Characteristic polynomialSimilar matrices

Statement

Let KK be a commutative ring, let AA be an n×nn \times n matrix over KK, and let PP be an invertible n×nn \times n matrix over KK. Then the characteristic polynomials of AA and P−1APP^{-1} A P are equal: pP−1AP(t)=pA(t).p_{P^{-1} A P}(t) = p_A(t). So similar matrices have the same characteristic polynomial.

Remarks

The proof is det⁡(tI−P−1AP)=det⁡(P−1(tI−A)P)=det⁡(tI−A)\det(t I - P^{-1} A P) = \det(P^{-1} (t I - A) P) = \det(t I - A). Hence the characteristic polynomial of a linear map does not depend on the basis. The converse fails: I2I_2 and (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} both have p(t)=(t−1)2p(t) = (t - 1)^2, but they are not similar.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionSimilar matrices, definition, no evidence levelSimilar matricesdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionCharacteristic polynomial does not depend on the basis, statement, no evidence levelCharacteristic polyno…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.
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Matrix multiplication definition
  5. Square matrix definition
  6. Linear map definition
  7. Invertible matrix definition
  8. Multilinear map definition
  9. Similar matrices definition
  10. Alternating multilinear map definition
  11. Determinant definition
  12. Characteristic polynomial definition
  13. Characteristic polynomial does not depend on the basis statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Properties
About this page
Tag0TYMASG33Y
URIm2://0TYMASG33Y
Kindstatement, proposition
WritingW0 stub
MSC15A15, 15A18
StatusThis copy of the site has no build data. Run m2 status 0TYMASG33Y.
UsesCharacteristic polynomial, Similar matrices

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