3.3 Homog. Auton. Linear Systems of Eqn

Question 1

Auton, homog, linear systems take the form (in vector)

Answer 1

x’ = Ax

Question 2

Def of eigenvalue:

Answer 2

There is a nonzero vector st. Av = lmda * v. We say that the v and lambda are an eigenvector/eigenvalue pair

Question 3

Invertible matrix thm, following 3 are equivalent:

Answer 3

1. Det != 0
2. Matrix is invertible,
3. Ax has only the zero soln x = 0 vector.

Question 4

How to solve for eigenvalues?

Answer 4

when det(A - lmda* I) = 0 AKA A minus lambda times the identity matrix

Question 5

How to find eigenvectors corresponding to lmda1, lmda2?

Answer 5

Find v when (A - lmda * I)v = 0.
AKA Matrix times vector v equal 0, solve for each component of the vector, there should be one degree of freedom

Question 6

If lamba, v is a eigenpair, then what is the solution to x’ = Ax

Answer 6

x = v * e^ (lmda*t) AKA c1v1e^(lmda1 * t) +
c2v2e^(lmda2 * t)

Question 7

What to do if you get imaginary eigenvalues to x’ = Ax?

Answer 7

Equation should be solved as normal for one imaginary soln. Then use eulers’ to turn it into cos,sin. Then, split the solns into imaginary and real. Then, those two are your solns, take off the i and those are your two solns

Question 8

Equation for e^ai*t in terms of cos, sin?

Answer 8

cos(at) + i sin(at)

Question 9

What happens if there is a multiplicity on an eigenvalue? What do you try first?

Answer 9

Solve normally, if it creates 2 different vectors then all is kosher. If not, then it gets more complex.

Question 10

What happens if there is multiplicity on eigenvalues and only correspond to the one eigenvector?

Answer 10

x1 = v * e^(lmda * t)
x2 = ( w + tv ) * e^(lmda * t)
Where v ,lmda is the eigenpair, and w is the generalized eigenvector.

Question 11

How to get the generalized eigenvector w?

Answer 11

(A - lmda * I) * w = v where v is the eigenvector.