Math 2.0

Linear and multilinear algebra; matrix theory Canonical forms and decompositions Theorem

Simultaneous diagonalisation

Builds on Diagonalisable matrix

Statement

Let FF be a field and let AA and BB be diagonalisable n×nn \times n matrices over FF. Then there is one invertible n×nn \times n matrix PP such that both P−1APP^{-1} A P and P−1BPP^{-1} B P are diagonal if and only if AB=BAAB = BA.

Remarks

Diagonal matrices commute, so the condition AB=BAAB = BA is necessary. For sufficiency, BB maps each eigenspace of AA to itself, and it is diagonalisable on each of them. The same holds for each family of pairwise commuting diagonalisable matrices.

Reading path

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionDiagonal matrix, definition, no evidence levelDiagonal matrixdefinitionSimilar matrices, definition, no evidence levelSimilar matricesdefinitionDiagonalisable matrix, definition, no evidence levelDiagonalisable matrixdefinitionSimultaneous diagonalisation, statement, no evidence levelSimultaneous diagonal…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. Matrix multiplication definition
  4. Square matrix definition
  5. Invertible matrix definition
  6. Diagonal matrix definition
  7. Similar matrices definition
  8. Diagonalisable matrix definition
  9. Simultaneous diagonalisation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Diagonalizable_matrix, Simultaneous diagonalization
About this page
Tag0FK17N4DKA
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Kindstatement, theorem
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MSC15A20, 15A27
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UsesDiagonalisable matrix

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