Math 2.0

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

Triangularisation

Builds on Characteristic polynomialUpper triangular matrixSimilar matrices

Statement

Let FF be a field and let AA be an n×nn \times n matrix over FF. Then AA is similar to an upper triangular matrix over FF if and only if the characteristic polynomial of AA splits over FF. Here, to split means to be a product of linear factors. In particular, over an algebraically closed field, such as ℂ\C, each square matrix is similar to an upper triangular matrix.

Remarks

The diagonal entries of the triangular matrix are the eigenvalues of AA, with their algebraic multiplicities. The proof is by induction: choose an eigenvector, extend it to a basis, and work on the quotient space. Over ℂ\C, the matrix PP can even be chosen unitary; this is the Schur decomposition.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

Show the path as a graph
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 matrixdefinitionUpper triangular matrix, definition, no evidence levelUpper triangular matr…definitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionSimilar matrices, definition, no evidence levelSimilar matricesdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionTriangularisation, statement, no evidence levelTriangularisationstatement
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 map definition
  7. Invertible matrix definition
  8. Upper triangular matrix definition
  9. Multilinear map definition
  10. Similar matrices definition
  11. Alternating multilinear map definition
  12. Determinant definition
  13. Characteristic polynomial definition
  14. Triangularisation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Triangular_matrix, Triangularisability
About this page
Tag0363JMQ42Y
URIm2://0363JMQ42Y
Kindstatement, theorem
WritingW0 stub
MSC15A20, 15A21
StatusThis copy of the site has no build data. Run m2 status 0363JMQ42Y.
UsesCharacteristic polynomial, Upper triangular matrix, Similar matrices

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