Eigenpair of affine transformation
Dependencies:
If $(\lambda, v)$ is an eigenpair of matrix $A$, then $(aλ+b, v)$ is an eigenpair of $aA+bI$.
Proof
$(\lambda, v)$ is eigenpair of $A \iff Av = \lambda v$. \[ (aA+bI)v = a(Av) + b(Iv) = a(\lambda v) + bv = (a\lambda + b)v \]
Dependency for:
Info:
- Depth: 12
- Number of transitive dependencies: 49
Transitive dependencies:
- /linear-algebra/vector-spaces/condition-for-subspace
- /linear-algebra/matrices/gauss-jordan-algo
- /sets-and-relations/composition-of-bijections-is-a-bijection
- /sets-and-relations/equivalence-relation
- Group
- Ring
- Polynomial
- Integral Domain
- Comparing coefficients of a polynomial with disjoint variables
- Field
- Vector Space
- Linear independence
- Span
- Linear transformation
- Composition of linear transformations
- Vector space isomorphism is an equivalence relation
- Semiring
- Matrix
- Stacking
- System of linear equations
- Product of stacked matrices
- Matrix multiplication is associative
- Reduced Row Echelon Form (RREF)
- Matrices over a field form a vector space
- Row space
- Elementary row operation
- Every elementary row operation has a unique inverse
- Row equivalence of matrices
- Row equivalent matrices have the same row space
- RREF is unique
- Identity matrix
- Inverse of a matrix
- Inverse of product
- Elementary row operation is matrix pre-multiplication
- Row equivalence matrix
- Equations with row equivalent matrices have the same solution set
- Basis of a vector space
- Linearly independent set is not bigger than a span
- Homogeneous linear equations with more variables than equations
- Rank of a homogenous system of linear equations
- Rank of a matrix
- Basis of F^n
- Matrix of linear transformation
- Coordinatization over a basis
- Basis changer
- Basis change is an isomorphic linear transformation
- Vector spaces are isomorphic iff their dimensions are same
- Canonical decomposition of a linear transformation
- Eigenvalues and Eigenvectors