Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Trace is a coefficient of the characteristic polynomial

Builds on Characteristic polynomialTrace of a matrix

Statement

Let KK be a commutative ring, let n≥1n \ge 1, and let AA be an n×nn \times n matrix over KK. Then the coefficient of tn−1t^{n-1} in the characteristic polynomial pA(t)=det⁡(tIn−A)p_A(t) = \det(t I_n - A) is −tr⁡A-\tr A, where tr⁡\tr is the trace. So pA(t)=tn−(tr⁡A) tn−1+⋯p_A(t) = t^n - (\tr A)\, t^{n-1} + \cdots.

Remarks

In the Leibniz expansion of det⁡(tIn−A)\det(t I_n - A), only the product of the diagonal entries (t−a11)⋯(t−ann)(t - a_{11}) \cdots (t - a_{nn}) contributes to tn−1t^{n-1}. For n=2n = 2, pA(t)=t2−(tr⁡A)t+det⁡Ap_A(t) = t^2 - (\tr A) t + \det A. Together with the constant coefficient, it shows that tr⁡\tr and det⁡\det are invariants of similarity.

Reading path

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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 mapdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionTrace is a coefficient of the characteristic polynomial, statement, no evidence levelTrace is a coefficien…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. Trace of a matrix definition
  7. Multilinear map definition
  8. Alternating multilinear map definition
  9. Determinant definition
  10. Characteristic polynomial definition
  11. Trace is a coefficient of the characteristic polynomial statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Properties
About this page
Tag0FRYPZ6D0X
URIm2://0FRYPZ6D0X
Kindstatement, proposition
WritingW0 stub
MSC15A15, 15A18
StatusThis copy of the site has no build data. Run m2 status 0FRYPZ6D0X.
UsesCharacteristic polynomial, Trace of a matrix

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