what is eigen values and eigen vectors?
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If you get nothing out of this quick review of linear algebra you must get this section. Without this section you will not be able to do any of the differential equations work that is in this chapter.
So let’s start with the following. If we multiply an n x n matrix by an n x 1 vector we will get a new n x 1 vector back.
What we want to know is if it is possible for the following to happen. Instead of just getting a brand new vector out of the multiplication is it possible instead.
to have matrix multiplication be the same as just multiplying the vector by a constant? Of course, we probably wouldn’t be talking about this if the answer was no. So, it is possible for this to happen, however, it won’t happen for just any value of λ . If we do happen to have a λ for which this works (and they will always come in pairs) then we call λ an eigenvalue of A and an eigenvector of A.
So let’s start with the following. If we multiply an n x n matrix by an n x 1 vector we will get a new n x 1 vector back.
What we want to know is if it is possible for the following to happen. Instead of just getting a brand new vector out of the multiplication is it possible instead.
to have matrix multiplication be the same as just multiplying the vector by a constant? Of course, we probably wouldn’t be talking about this if the answer was no. So, it is possible for this to happen, however, it won’t happen for just any value of λ . If we do happen to have a λ for which this works (and they will always come in pairs) then we call λ an eigenvalue of A and an eigenvector of A.
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