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Jordan form over the base field exists exactly when the characteristic polynomial splits
Statement
An endomorphism of a finite-dimensional -vector space has a Jordan canonical form over if and only if its characteristic polynomial splits into linear factors over . This includes the zero space, where the characteristic polynomial is and the Jordan form is empty.
Facts & Assumptions
Given: An endomorphism of a finite-dimensional -vector space .
If splits, is the direct sum of invariant generalised eigenspaces , and is nilpotent (Split characteristic polynomials decompose into generalised eigenspaces of the algebraic multiplicities).
Every finite-dimensional nilpotent endomorphism has a basis of Jordan strings (Every finite-dimensional nilpotent endomorphism has a basis of Jordan strings).
The characteristic polynomial of a block triangular matrix is the product of the characteristic polynomials of its diagonal blocks, including empty blocks (The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks).
Proof
Suppose splits. For every from [L1], apply [L2] to ; its nilpotent strings are Jordan strings for at , and concatenating their bases across the direct sum gives a Jordan basis of .
Conversely, if has a Jordan basis, its matrix is block diagonal with blocks for ; [L3] gives , which splits over .
Steps 1.1 and 1.2 prove both directions; the empty direct sum and empty product prove the zero-space case.
Depends on
- Jordan bases and Jordan canonical forms over the base field
- Split characteristic polynomials decompose into generalised eigenspaces of the algebraic multiplicities
- Every finite-dimensional nilpotent endomorphism has a basis of Jordan strings
- The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks
Used by
- Every finite-dimensional endomorphism over an algebraically closed field has Jordan form Corollary
- Algebraic multiplicity of a nonzero compact-operator eigenvalue Definition
- The real quarter-turn acquires diagonal Jordan form over ℂ Example
- The real quarter-turn has rational canonical form but no real Jordan form Example
- FALSE: Every finite-dimensional endomorphism has Jordan form over its base field False statement
- Over a perfect field, every endomorphism has a unique commuting semisimple-plus-nilpotent decomposition, polynomial in the endomorphism Theorem
- Split matrices are similar exactly when their Jordan block multisets agree Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- S. Axler, Linear Algebra Done Right, 4th ed., Results 8.42-8.46 (standard reference, not scraped)
- S. Treil, Linear Algebra Done Wrong, Chapter 9, Section 5 (standard reference, not scraped)