Eigenvector and Eigenvalue Flashcards
What is an eigenvalue?
T = linear operator on V
λ = eigenvalue if there exists a non-zero vector such that T(v) = λv
What are the properties of the linear map
λid(v) - T?
- Not injective
- Not surjective
- Not invertable
When is a linear operator invertible?
det(T) != 0
det(T) := determinant of matrix of T relative to ANY BASIS of V
What is the characteristic polynomial?
XT of linear operator on finite dimensional VS
XT(λ):= det(λidv - T)
What are the 3 equivalent definitions for eigenvalue?
- λ is eigenvalue
- λidv - T is not invertible
- XT(λ) = 0
What is the eigenvector of a linear operator?
Nonzero vector such that T(v) = λv for some eigenvalue.
What is the eigenspace?
Eigenspace of T corresponding to λ is the subspace of V such that T(v) = λv and
What is the other notation for eigenspace and why?
Eλ = Ker(λid - T) since T(v) = λ is equivalent to (λid - T)(v) = 0.
Write as parametric form if need be.
Why is an eigenspace one-dimensional when eigenvalues are distinct?
- Eigenspace is set of all vectors corresponding to λ
- When λ distinct, each λ has one vector corresponding to it.
- Since each λ distinct, the only scalar multiple of vector vi corresponding to λi is c . vi
What are all the properties when eigenvalues are distinct?
- Each eigenspace is strictly one dimensional
- ## Every vector is independent
What are the eigenvalues of a nxn matrix A?
det(λ I - A) = 0
What is the algebraic and geometric multiplicities of an eigenvalue?
- Algebraic: Number of times eigenvalue appears as root of Char Poly.
- Geometric: Dimensions of eigenspace corresponding to eigenvalue
How is the eigenspace calculated and its dimensions, calculated??
Kernel/Null space of lamba I - T
Dimensions = dimension of above null space = nullity of above null space =
