Alphabeta MathDiagonalisation and the Minimal Polynomial: Examples and Counterexamples
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✓ 12 results · all verified · 5 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Example
Minimal polynomials of scalar and diagonal endomorphisms, including the zero-dimensional case
Example
The identity and a nontrivial Jordan block have the same characteristic polynomial but different minimal polynomials
Example
Computing the minimal polynomial of an explicit 3×3 idempotent from its annihilating polynomials
Example
A nilpotent shift has minimal polynomial xⁿ and, for n>0, a single primary component
Example
Quarter-turn rotation is not diagonalisable over ℝ but is diagonalisable over ℂ
Example
Primary decomposition over ℚ with one linear and one irreducible quadratic factor
Example
Lagrange polynomials give the three eigenspace projections of a diagonalisable endomorphism
Example
Two commuting non-scalar matrices simultaneously diagonalised in one explicit basis
Counterexample
Commuting alone does not imply simultaneous diagonalisability
False statement
FALSE: If the minimal polynomial splits, then the endomorphism is diagonalisable
False statement
FALSE: The characteristic polynomial determines whether an endomorphism is diagonalisable
False statement
FALSE: A diagonalisable endomorphism must have a characteristic polynomial with distinct roots
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §5B
- Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §3
- Keith Conrad, The Minimal Polynomial and Some Applications, Example 4.2
- Keith Conrad, The Minimal Polynomial and Some Applications, Examples 4.6, 4.9, and 4.16
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §§8A–8B
- Keith Conrad, Potential Diagonalizability, Example 1
- Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §5, Theorem 5.19
- Keith Conrad, The Minimal Polynomial and Some Applications, Remark 4.12
- Keith Conrad, The Minimal Polynomial and Some Applications, Remark 5.3
- Keith Conrad, The Minimal Polynomial and Some Applications, Theorem 4.11
- Keith Conrad, The Minimal Polynomial and Some Applications, Example 4.2 and Theorem 4.11
- Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §3, examples and Theorem 5.14