Math 2.0

Linear and multilinear algebra; matrix theory Corollary

Distinct eigenvalues imply diagonalisable

Builds on A matrix is diagonalisable iff it has a basis of eigenvectorsEigenvectors with distinct eigenvalues are linearly independent

Statement

Let FF be a field and let AA be an n×nn \times n matrix over FF. If AA has nn pairwise distinct eigenvalues in FF, then AA is diagonalisable over FF.

Remarks

Eigenvectors for the nn distinct eigenvalues are linearly independent, so they form a basis of FnF^n. The converse fails: InI_n is diagonal, but it has only one eigenvalue. For example, (1203)\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} has eigenvalues 11 and 33, so it is diagonalisable.

Reading path

To understand this page from the ground up, read these 22 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 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…statementEigenvectors with distinct eigenvalues are linearly independent, statement, no evidence levelEigenvectors with dis…statementDistinct eigenvalues imply diagonalisable, statement, no evidence levelDistinct eigenvalues …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.
  • no evidence level
  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
  21. Eigenvectors with distinct eigenvalues are linearly independent statement
  22. Distinct eigenvalues imply diagonalisable statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Diagonalizable_matrix, Characterization
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MSC15A20
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UsesA matrix is diagonalisable iff it has a basis of eigenvectors, Eigenvectors with distinct eigenvalues are linearly independent

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