Alphabeta MathEigenvalues and the Characteristic Polynomial: Examples and Counterexamples
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✓ 7 results · all verified · 2 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- 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
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and 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
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
The matrix beginpmatrix2&10&3 endpmatrix has characteristic polynomial (x-2)(x-3) and two explicitly computed eigenspaces
Example
A quarter-turn of ℝ² has characteristic polynomial x²+1 and no real eigenvalue
Example
beginpmatrix0&11&1 endpmatrix over F₂ has characteristic polynomial x²+x+1 and no eigenvalue in its base field
Example
beginpmatrix2&10&2 endpmatrix has algebraic multiplicity two but geometric multiplicity one at 2
Example
The scalar matrix 2I₂ has algebraic and geometric multiplicity two at 2
Example
Cayley-Hamilton reduces every power of beginpmatrix1&11&0 endpmatrix to a linear combination of A and I
False statement