A non-zero vector is an eigenvector of if for some
Let over then the column vector is said to be an eigenvector of A if for some ,. is an eigenvector of The corresponding linear map
the is called the eigenvalue of A
Proposition
if is an eigenvalue of a linear map the is also an eigenvalue of the matrix of wrt some basis of
Let be a basis of
The matrix of wrt is a diagonal matrix ifff each of the basis elements all eigenvectors.
Matrix Version:
matrix A is similar to a diagonal matrix iff admits a basis consisting of eigenvectors.
If A is similar to a diagonal matrix then is a diagonal matrix and are the eigenvectors of are eigenvalues of if we choose as the basis of
other direction is trivial
is a eigenvalue of iff there exist a non-zero vector such that
if A is the associated matrix of wrt to some basis
so is singular ,, det
The equation is the characteristic polynomial of A (also ?)
and roots of which lies in are called the eigenvalues
Claim
Proposition
Let be distinct eigenvalues of and let be the corresponding eigenvectors of then is a linear independent set in
so we define
where A is a matrix associated with wrt some basis
The constant term of is the values at ,.
the coeff of is is -trace (A)
sum of eigenvalues is the trace of A and product is the
Theorem
Let be a linear operator on a finite dim vector space let be the distinct eigenvalues of let be the eigenspace of
TFAE