An eigenvector of A is a nonzero vector v in Rn such that Av = λv for some scalar λ
An eigenvalue is a scalar λ such that the equation Av=λv has a nontrivial solution
Eigenvalue can be zero, eigenvector cannot
Eigenspace is the vector space formed by all eigenvectors corresponding to a specific eigenvalue, along with 0 vector, you can say eigenspace is the basis for n eigenvectors in Rn
Find the solution set of ( A − λ In )v = 0, or the null space of A − λ In
Steps to find eigenvalues and eigenvectors
Characteristics polynomial: f ( λ )= det ( A − λ I n ) .
Factor the polynomial by using quadratic formula or some special magic to find eigenvalues
Plug eigenvalues back to the Nul( A − λ In ) = 0 to find eigenvectors
Diagonalization
An nxn matrix is diagonalizable if A = CDC-1
nxn matrix is diagonalizable iff A has n linearly independent eigenvectors
Algebraic and Geometric Multiplicity
Algebraic: is the # of the root per root
Geometric: dimension of eigenspace/null space/# of free variables
