Eigenvalues & Eigenvectors Flashcards
What is the set of eigenvalues called?
The spectrum.
What is the eigenspace of an eigenvalue?
The set of eigenvectors corresponding to an eigenvalue.
What is the characteristic equation for finding eigenvalues?
(A - lambdaI)x = 0.
When will the system (A - lambdaI)x have non-zero solutions in x?
Iff it is indeterminate, i.e. r < n.
This means det(A - lambda*I) = 0.
For an n*n matrix, how many eigenvalues can you have?
n or fewer eigenvalues.
How many eigenvectors will each eigenvalue have?
1 or more eigenvectors.
What can be said about the different eigenvectors?
That they are mutually independent.
What is the algebraic order of an eigenvalue (M) ?
The power of the root of the characteristic polynomial.
What is the sum of the algebraic orders?
n
What is the geometric order (m) ?
The dimension of the eigenspace of an eigenvalue, i.e. the space spanned by the eigenvectors.
What is the relationship between M and m?
m < or = M
How to check the eigenvalues?
- The product of the eigenvalues with their algebraic orders as powers = det(A).
- The sum of the eigenvalues multiplied by their algebraic orders = sum of diagonal terms of A = trace(A).
