Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Diagonalisable iff the minimal polynomial has distinct roots

Builds on Diagonalisable matrixMinimal polynomial

Statement

Let FF be a field and let AA be an n×nn \times n matrix over FF with minimal polynomial mAm_A. Then AA is diagonalisable over FF if and only if mA(t)=(t−λ1)(t−λ2)⋯(t−λk)m_A(t) = (t - \lambda_1)(t - \lambda_2) \cdots (t - \lambda_k) for some pairwise distinct λ1,…,λk∈F\lambda_1, \dots, \lambda_k \in F.

Remarks

In words: mAm_A splits over FF and has no repeated roots. If A2=AA^2 = A, then mAm_A divides t2−t=t(t−1)t^2 - t = t(t - 1), so each projection matrix is diagonalisable. The matrix (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} has mA=(t−1)2m_A = (t - 1)^2, so it is not diagonalisable.

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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 matrixdefinitionDiagonal matrix, definition, no evidence levelDiagonal matrixdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSimilar matrices, definition, no evidence levelSimilar matricesdefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionDiagonalisable matrix, definition, no evidence levelDiagonalisable matrixdefinitionAnnihilating polynomial, definition, no evidence levelAnnihilating polynomi…definitionMinimal polynomial, definition, no evidence levelMinimal polynomialdefinitionDiagonalisable iff the minimal polynomial has distinct roots, statement, no evidence levelDiagonalisable iff th…statement
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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. Diagonal matrix definition
  9. Endomorphism definition
  10. Similar matrices definition
  11. Polynomial of an endomorphism definition
  12. Diagonalisable matrix definition
  13. Annihilating polynomial definition
  14. Minimal polynomial definition
  15. Diagonalisable iff the minimal polynomial has distinct roots statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Diagonalizable_matrix, Characterization
About this page
Tag0MPAJRB23T
URIm2://0MPAJRB23T
Kindstatement, theorem
WritingW0 stub
MSC15A20
StatusThis copy of the site has no build data. Run m2 status 0MPAJRB23T.
UsesDiagonalisable matrix, Minimal polynomial

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