Karteikarten
Basic definitions
Karten in diesem Stapel (15)
- 1Definition: Eigenvector Eigenvalue EigenpairA"x"= λ"x"
A square
x =/= 0
λ is eigenvalue
x is eigenvector
(λ,x) is an eigenpair - 2Characteristic polynomialp(λ)=det(λI-A)
degree n - 3Spectrum of AThe set of all eigenvalues of A
Λ(A)={λ1, λ2, ...,λn} - 4Algebraic multiplicityMultiplicity of λ as a zero in p(λ)
i.e.
if (λ-x)^k=0
then λ=x has a algebraic multiplicity of k - 5Invariant subspaceX is invariant if:
AX is subset of X
i.e. if x in X then so is Ax.
or equiv.
AX = XB for some b
then range(X) is invariant for A - 6Deduce what you can from AX = XB1. Range(X) is a invariant subspace of A
2. Spectrum of B (Λ(B)) is a subset of Λ(A) - 7A and B are similarB=inv(P)AP,
P non-singular
A,B squre, same dimensions - 8Transforming matrixIn the equation
B=inv(P)AP,
P is a transforming matrix - 9similarity transformationB=inv(P)AP,
P non-singular
A,B squre, same dimensions - 10A and B are similar., (B=inv(P)AP) What is true of their eigenvalues and vectors?1. A and B have the same eigenvalues (Λ(A) = Λ(B))
2. (λ, x) is an eigenpair of A iff (λ,inv(P)x) is an eigenpair of B - 11A and B are similar., (A=inv(P)BP) What is true of their eigenvalues and vectors?1. A and B have the same eigenvalues (Λ(A) = Λ(B))
2. (λ, x) is an eigenpair of B iff (λ,inv(P)x) is an eigenpair of A - 12Unitary similarB=U*AU,
U is unitary - 13Orthogonally similarB = Q^TAQ,
Q orthogonal - 14A is simpleA is diagonizable:
A is similar to a diagonal matrix,
A = inv(P)DP,
D is diagonal - 15A is diagonizableA is simple:
A is similar to a diagonal matrix,
A = inv(P)DP,
D is diagonal
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- Beschreibung
- Matrix Analysis (Theory of eigensystems) Karteikarten am Basic definitions, erstellt von Simon Majgaard am 08/11/2020.
- Erstellt von
- Simon Majgaard
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