Alphabeta MathDiagonalisation and the Minimal Polynomial
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✓ 22 results · all verified · 16 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
- Group Homomorphisms and the Isomorphism Theorems
- 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
- 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
Definition
A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation
Definition
The annihilator set Ann(T)={p∈ F[x]:p(T)=0}; once existence is proved, its unique monic generator μ_T is the minimal polynomial
Theorem
The annihilator ideal is nonzero and has a unique monic generator; p(T)=0 if and only if μ_T∣ p
Theorem
The minimal polynomial is unchanged by choosing a matrix representation or replacing a matrix by a similar one
Lemma
For a matrix over a field, extending the scalar field does not change its minimal polynomial
Corollary
The minimal polynomial divides the characteristic polynomial, μ_T∣χ_T
Theorem
Over every extension field, a scalar is an eigenvalue of the extended matrix exactly when it is a root of the minimal polynomial
Theorem
The minimal and characteristic polynomials have exactly the same monic irreducible factors
Lemma
For every polynomial p, both ker p(T) and imp(T) are T-invariant
Proposition
The minimal polynomial of a restriction to an invariant subspace divides the original minimal polynomial
Lemma
If gcd(f,g)=1 and (fg)(T)=0, then V=ker f(T)⊕ker g(T)
Definition
Primary components ker q(T)ᵉ and generalised eigenspaces G_λ⁽ᵉ⁾(T)=ker(T-λ I)ᵉ
Theorem
Primary decomposition: the irreducible-power factors of μ_T split V into their invariant kernels
Corollary
Each projection in the primary decomposition is a polynomial in the endomorphism
Corollary
If the minimal polynomial splits, V is the direct sum of the stabilised generalised eigenspaces
Theorem
An endomorphism is diagonalisable exactly when V=⨁_i<rE_λᵢ(T) for some finite list of distinct scalars λᵢ
Theorem
An endomorphism is diagonalisable if and only if its characteristic polynomial splits and every eigenvalue's geometric multiplicity equals its algebraic multiplicity
Theorem
An endomorphism is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors
Corollary
A characteristic polynomial that splits into distinct linear factors forces diagonalisability
Corollary
Every idempotent endomorphism is diagonalisable and is projection onto its image along its kernel
Theorem
The restriction of a diagonalisable endomorphism to an invariant subspace is diagonalisable
Definition
Simultaneous diagonalisability: one basis that diagonalises every endomorphism in a family
Lemma
Commuting endomorphisms preserve each other's eigenspaces
Theorem
A family of diagonalisable endomorphisms of a finite-dimensional space is simultaneously diagonalisable if and only if its members commute pairwise
Corollary
If two commuting endomorphisms are diagonalisable, then every finite linear combination of products of their powers is diagonalisable; in particular, their sum and product are diagonalisable
Corollary
The algebra F[T] generated by an endomorphism is isomorphic to F[x]/(μ_T)
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §§5A–5B
- Keith Conrad, The Minimal Polynomial and Some Applications, §§4–5
- Keith Conrad, The Minimal Polynomial and Some Applications, §4
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §5B
- Keith Conrad, The Minimal Polynomial and Some Applications, §4, Theorem 4.4
- Keith Conrad, The Minimal Polynomial and Some Applications, §4, Theorem 4.3
- Keith Conrad, Potential Diagonalizability, Theorem 4(1)
- Keith Conrad, The Minimal Polynomial and Some Applications, Corollary 4.10
- Sheldon Axler, Linear Algebra Done Right, 4th ed., Theorem 8.30
- Keith Conrad, The Minimal Polynomial and Some Applications, Theorem 4.7
- Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §5
- Keith Conrad, The Minimal Polynomial and Some Applications, §5, Lemma 5.1
- Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §5, proof of Theorem 5.19
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §§8A–8B
- Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §5, Theorem 5.19
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §8B, Theorem 8.22 (complex-field special case)
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §5D
- Keith Conrad, The Minimal Polynomial and Some Applications, Theorem 4.11
- Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §3, Theorem 5.14
- Keith Conrad, The Minimal Polynomial and Some Applications, Corollary 4.14
- Keith Conrad, The Minimal Polynomial and Some Applications, §5
- Keith Conrad, The Minimal Polynomial and Some Applications, Theorem 5.2 and Corollary 5.4
- Keith Conrad, The Minimal Polynomial and Some Applications, Corollary 5.5
- Keith Conrad, The Minimal Polynomial and Some Applications, closing observation in §4