Math 2.0

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

Rational canonical form

Builds on Companion matrixSimilar matricesBlock diagonal matrix

Statement

Let FF be a field and let AA be an n×nn \times n matrix over FF. Then there are unique monic nonconstant polynomials f1,…,fr∈F[t]f_1, \dots, f_r \in F[t] with f1∣f2∣⋯∣frf_1 \mid f_2 \mid \cdots \mid f_r and the following property. The matrix AA is similar to the block diagonal matrix (C(f1)⋱C(fr)),\begin{pmatrix} C(f_1) & & \\ & \ddots & \\ & & C(f_r) \end{pmatrix}, where C(fi)C(f_i) is the companion matrix of fif_i. This block matrix is the rational canonical form of AA.

Remarks

The polynomials f1,…,frf_1, \dots, f_r are the invariant factors of AA. The last factor frf_r is the minimal polynomial of AA, and f1f2⋯frf_1 f_2 \cdots f_r is its characteristic polynomial. Unlike the Jordan form, it needs no field extension: it works over any field. Two matrices over FF are similar if and only if they have the same invariant factors.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionBlock matrix, definition, no evidence levelBlock matrixdefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionBlock diagonal matrix, definition, no evidence levelBlock diagonal matrixdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionMultilinear 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…definitionCompanion matrix, definition, no evidence levelCompanion matrixdefinitionRational canonical form, statement, no evidence levelRational canonical fo…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. Vector space definition
  4. Block matrix definition
  5. Matrix multiplication definition
  6. Square matrix definition
  7. Linear map definition
  8. Block diagonal matrix definition
  9. Invertible matrix definition
  10. Multilinear map definition
  11. Similar matrices definition
  12. Alternating multilinear map definition
  13. Determinant definition
  14. Characteristic polynomial definition
  15. Companion matrix definition
  16. Rational canonical form statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Frobenius_normal_form, General case and theory
About this page
Tag042JWAR3Q0
URIm2://042JWAR3Q0
Aliasesfrobenius-normal-form
Kindstatement, theorem
WritingW1 stated
MSC15A21
StatusThis copy of the site has no build data. Run m2 status 042JWAR3Q0.
UsesCompanion matrix, Similar matrices, Block diagonal matrix
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