Eigenvalues

Question 1

What is an eigenvalue

Answer 1

An eigenvalues is a L s.t phi(v) = Lv

Question 2

What is L-eigenspace

Answer 2

L-eigenspace is set of all eigenvalues associated to the E.vectors, ker(phi - LI) consists of all eigenvectors of phi with 0
this is a vector subspace because it’s a kernel of a linear operaor

Question 3

What is characteristic polynomial

Answer 3

Characteristic polynomial is det(phi - LI)

Question 4

What is det of linear map

Answer 4

Det of linear map is det of matrix B representing on phi

Question 5

What is algebraic multiplicity

Answer 5

Algebraic multiplicity is no of times L is repeated as root of characteristic polynomial

Question 6

What is geometric multiplicity

Answer 6

Geometric multiplicity is dim E(L) dimension of L-eigenspace, no of L.I vectors associated with E,value

Question 7

With phi(v) = Lv, then what is p(phi)(v)

Answer 7

If phi(v) = Lv, then p(phi)(v) = p(L)v L is eigenvalue and v is eigenvector

Question 8

If phi is a linear operator, what is any eigenvalue of phi

Answer 8

If phi is a linear operator, any eigenvalue of hi is a root of M(phi) (because of Cayley Hamilton theorem)

Question 9

What is Cayley Hamilton theorem

Answer 9

Cayley Hamilton theorem is:
Any matrix/linear operator satisfies it’s characteristic polynomial, if input matrix into characteristic polynomial result will be 0
See E.g in lecture notes

Question 10

What does minimum polynomial of phi divide and what are roots of minumum polynomial

Answer 10

Minimum polynomial of phi divide characteristic polynomial of phi, same for matrices and roots of minimum polynomial are eigenvalues of phi (as they are roots of characteristic polynomial)