Question

If matrix A has an inverse A^-1, use equation 2: Av = lambda*v, to show that A^-1 has the same eigenvectors as A. Determine a relationship between the eigenvalues of A and A^-1. Illustrate with a suitable example.

Answer 1

what happens if you multiply both sides by A^-1 ?

Answer 2

Then eventually you'd get A^-1 = 1/lambda, but there was another problem like this and this was the answer to it, but this one is a little different, so u know what to do here?

Answer 3

you start with
\[ Av = \lambda v\]
multiply both sides by A inverse:
\[ A^{-1}Av = A^{-1} \lambda v\]

what do you get on the left side?
what do you get on the right side?

Answer 4

On the left side, you get just v and on the right side you get A^-1 lambda v

Answer 5

yes, and remember A^-1 is a matrix, lambda is a scalar (a simple number) and v is a vector. when you multiply a number times a matrix, you can switch the order, so you can write it as:
\[ v= \lambda A^{-1}v \]
now divide both sides by lambda (assuming it is not zero)
\[ \frac{1}{\lambda} v = A^{-1}v \]
or
\[ A^{-1}v = \frac{1}{\lambda} v \]

Does that look familiar? Matrix times a vector = a number times that same vector

Answer 6

So then does that answer our question? And what would be a suitable example like it tells us to do?

Answer 7

if you see
Av = lambda*v

then you know that v is an eigenvector of matrix A and lambda is an eigenvalue associated with eigenvector v

So what does
\[ A^{-1}v = \frac{1}{\lambda} v \]
tell you?

Answer 8

v is the ____ of matrix A^-1 and 1/lambda is the _____ associated with v

Answer 9

So v is the eigenvector and 1/lambda is the eigenvalue? Is this right?

Answer 10

yes. This shows that A and its inverse have the same eigenvectors, and their eigenvalues are the reciprocal (1/lambda and lambda)

Answer 11

So the answer would be that A and A^-1 are reciprocals of each other, would this be our answer?

Answer 12

So the answer would be that **the eigenvalues of *** A and A^-1 are reciprocals of each other, would this be our answer?

Answer 13

is that what you mean?

Answer 14

yea