Here's the question you clicked on:
Cali_Native559
Suppose that delta is an eigenvalue of an invertible matrix A. Show that 1/delta is an eigenvalue of A inverse.
write the definition of an eigenvalue
So I know what an eigenvalue is, but how would you write the definition of it, what do u mean by that?
Start with showing what an eigenvalue is: a scalar, \(\delta\) in this case, such that\[AI=\delta A\]Now, since we can assume \(A\) has an inverse \(A^{-1}\), multiply both sides by the inverse. What do you get?
*I meant of course\[AI=\delta I\]multiply both sides by \(A^{-1}\) and what do you get?
Thanks for the help
So I do what u said and that should be the answer? That should prove what we need to prove?
Not quite, you have not demonstrated what we have set out to prove. why don't you write out what you get by multiplying the definition of the eigenvalue\[AI=\delta I\]by the inverse of the matrix, \(A^{-1}\) and show me what you get? I will help you from there.
So this is what I did, so here's my work: A^-1 AI = delta*I*A^-1 I = delta*I*A^-1 So I that's what I got, I'm pretty sure the right side can't be simplified any further, so if you could help me with the rest, that'd be great.
divide both sides by delta and you're done check it out, you get the definition of the eigenvalue, with the eigenvalue of A^-1 as 1/delta
So on the right side, we then get I * A^-1, but that's just still A^-1 right?
yes\[AI=\delta I\]\[A^{-1}AI=II=I^2=I=A^{-1}\delta I\]remember that scalars are commutative so we can move delta around\[\delta A^{-1}I=I\]\[A^{-1}I=\frac1\delta I\]which is the definition of the eigenvalue
And one more thing, we also divide I from both sides too right, because that's how we get the one right?
I is the identity matrix, multiplying or dividing by it changes nothing, just like the scalar number 1
Right, so that'll just give us that one.
Because it's just I / I.
matrices do not become scalars matrix division is defined as multiplication by its inverse the inverse of the identity matrix is still the identity matrix, so\[II^{-1}=II=I^2=I\]still the identity matrix, not the scalar number 1 big difference, gotta get that straight in linear algebra
Ok, so I think I see, ok so in the end we get this: A^-1 * I = 1/delta * I, so then the I's just cancel out since they're on both sides and that is what gives us A^-1 = 1/delta right?
you don't need to cancel the I's, it's just like times 1 if I had 1x=1y would I need to cancel the 1's ? a matrix times the identity matrix is itself, so\[AI=AI^{-1}=A\] in the definition of the eigenvalue, I is written explicitly to show that the scalar eigenvalue lambda is multiplying into a matrix\[A=\lambda I\]. if the I's cancel you get\[A=\lambda\]that's wrong because the thing on the right is a scalar and the thing on the left is not Just leave your last line as\[A^{-1}=\frac1\delta I\]and that makes clear the properties of the eigenvalue we were looking for
All right that makes sense, and can u help out with one more problem related to this same stuff?
I can try, but my connection is horrible right now, I may only be able to pm you
Ok, so here's the 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.