Math 2.0

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

Jordan normal form

Builds on Jordan blockSimilar matricesBlock diagonal matrix

Statement

Let FF be an algebraically closed field, for example ℂ\C, and let AA be an n×nn \times n matrix over FF. Then AA is similar to a block diagonal matrix J=(Jk1(λ1)⋱Jkr(λr)),J = \begin{pmatrix} J_{k_1}(\lambda_1) & & \\ & \ddots & \\ & & J_{k_r}(\lambda_r) \end{pmatrix}, whose diagonal blocks are Jordan blocks. The matrix JJ is unique up to the order of the blocks. It is the Jordan normal form of AA.

Remarks

The λi\lambda_i are the eigenvalues of AA; they need not be distinct. The number of blocks for λ\lambda is its geometric multiplicity, and the sum of their sizes is its algebraic multiplicity. The largest block for λ\lambda has size equal to the multiplicity of λ\lambda in the minimal polynomial. The theorem holds over any field FF for the matrices whose characteristic polynomial splits over FF.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionBlock matrix, definition, no evidence levelBlock matrixdefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionBlock diagonal matrix, definition, no evidence levelBlock diagonal matrixdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionJordan block, definition, no evidence levelJordan blockdefinitionSimilar matrices, definition, no evidence levelSimilar matricesdefinitionJordan normal form, statement, no evidence levelJordan normal formstatement
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. Block matrix definition
  4. Matrix multiplication definition
  5. Square matrix definition
  6. Block diagonal matrix definition
  7. Invertible matrix definition
  8. Jordan block definition
  9. Similar matrices definition
  10. Jordan normal form statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Jordan_normal_form, Overview
About this page
Tag085HQ5G5TE
URIm2://085HQ5G5TE
Aliasesjordan-canonical-form
Kindstatement, theorem
WritingW1 stated
MSC15A21, 15A20
StatusThis copy of the site has no build data. Run m2 status 085HQ5G5TE.
UsesJordan block, Similar matrices, Block diagonal matrix
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