Math 2.0

Linear and multilinear algebra; matrix theory Theorem

A matrix is diagonalisable iff it has a basis of eigenvectors

Builds on Diagonalisable matrixEigenvectorBasis

Statement

Let FF be a field and let AA be an n×nn \times n matrix over FF. Then AA is diagonalisable if and only if FnF^n has a basis of eigenvectors of AA. If v1,…,vnv_1, \dots, v_n is such a basis with Avi=λiviA v_i = \lambda_i v_i, and PP has the columns v1,…,vnv_1, \dots, v_n, then P−1APP^{-1} A P is the diagonal matrix with entries λ1,…,λn\lambda_1, \dots, \lambda_n.

Remarks

The equation AP=PDAP = PD says column by column that Avi=λiviA v_i = \lambda_i v_i. Equivalently, FnF^n is the direct sum of the eigenspaces of AA. Equivalently, the sum of the geometric multiplicities of the eigenvalues is nn.

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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 combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionDiagonal matrix, definition, no evidence levelDiagonal matrixdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionSimilar matrices, definition, no evidence levelSimilar matricesdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionDiagonalisable matrix, definition, no evidence levelDiagonalisable matrixdefinitionEigenvector, definition, no evidence levelEigenvectordefinitionBasis, definition, no evidence levelBasisdefinitionA matrix is diagonalisable iff it has a basis of eigenvectors, statement, no evidence levelA matrix is diagonali…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 combination definition
  7. Linear map definition
  8. Linear subspace definition
  9. Invertible matrix definition
  10. Diagonal matrix definition
  11. Linear independence definition
  12. Endomorphism definition
  13. Linear span definition
  14. Similar matrices definition
  15. Eigenvalue definition
  16. Spanning set definition
  17. Diagonalisable matrix definition
  18. Eigenvector definition
  19. Basis definition
  20. A matrix is diagonalisable iff it has a basis of eigenvectors statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Diagonalizable_matrix, Characterization
About this page
Tag056SW1E0AC
URIm2://056SW1E0AC
Kindstatement, theorem
WritingW0 stub
MSC15A20
StatusThis copy of the site has no build data. Run m2 status 056SW1E0AC.
UsesDiagonalisable matrix, Eigenvector, Basis

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