Eigenvalues And Characteristic Polynomials

Question 1

What are eigenvectors and eigenvalues

Answer 1

Eigenvectors are vectors that remain on their own span during a linear transformation
Eigenvalues are the factor by which the vector is stretched/squished

Question 2

If matrix A is invertible, what are attributes of A

Answer 2

If matrix A is invertible, attributes of A are:
Phi(A) is an isomorphism
Col(1) to col(n) is L.I (by R-N-T)
Row(1) to row(n) is L.I (row rank = column rank = rank A)
Det =/ 0
RREF A = I

Question 3

What is an eigenvalue and eigenvector

Answer 3

An eigenvalue is a scalar L if there exists:
A * v = L*v where v is a vector n x 1
V is eigenvector of as with eigenvalue L (or L-eigenvalue)

Question 4

What is the definition of characteristic polynomial of A and what are eigenvalue s

Answer 4

definition of characteristic polynomial of A is:
Det(A-tI) where I is identity matrix
Eigenvalues are roots of characteristic polynomial

Question 5

What do similar matrices have

Answer 5

Similar matrices have:

Same characteristic polynomial and eigenvalues

Question 6

When is L an eigenvalue of linear operator phi

Answer 6

L is an eigenvalue of linear operator phi if:
Phi(v) = Lv and v =/ 0
v is a L-eigenvector of phi

Question 7

What is L-eigenspace of linear operator phi

Answer 7

L-eigenspace of linear operator phi is:
E(L) = all v in V s.t phi(v) = Lv = ker(phi - L id(V)) = solution space of MX = 0
Where M is A - Lid(V)

Question 8

What does L-eigenspace consist of

Answer 8

L-eigenspace consists of L-eigenvectors (all eigenvectors belonging to L) and 0

Question 9

What is the characteristic polynomial of a linear operator

Answer 9

Characteristic polynomial of a linear operator is:

Det(phi - t *id(V)) where V is a vector space