How do inverse matrices affect eigenvalues?

  • Thread starter Thread starter Old Guy
  • Start date Start date
  • Tags Tags
    Proof
Old Guy
Messages
101
Reaction score
1

Homework Statement


Given a matrix with eigenvalues \lambda_{i}, show that if the inverse of the matrix exists, its eigenvalues are \frac{1}{\lambda}.


Homework Equations





The Attempt at a Solution

This shouldn't be so hard. I've come up with a few trivial examples, but I would like to get a general proof in 3 (or more) dimensions. I've tried solving for the eigenvalues, getting the eigenvectors and trying a unitary transformation. I've tried substituting the product of the matrix with its inverse in the characteristic equation and playing around with that before actually calculating the determionant. I've tried other symbolic brute force attempts, but they get very complicated very quickly. I've researched a bunch of determinant identities to no avail. I feel that there must be some key relation that I'm just missing. Any help would be appreciated.
 
Physics news on Phys.org
Just write down the definition of eigenvalue and eigenvector, and then apply the inverse matrix to that equation. Nothing complicated involving determinants and/or unitary transformations are necessary.
 
Is this the idea?
$\begin{array}{l}<br /> \Lambda \left| \psi \right\rangle = \lambda \left| \psi \right\rangle \\ <br /> \Lambda ^{ - 1} \left( {\Lambda \left| \psi \right\rangle } \right) = \Lambda ^{ - 1} \left( {\lambda \left| \psi \right\rangle } \right) \\ <br /> \left( {\Lambda ^{ - 1} \Lambda } \right)\left| \psi \right\rangle = \left( {\Lambda ^{ - 1} \lambda } \right)\left| \psi \right\rangle \\ <br /> I\left| \psi \right\rangle = \lambda \Lambda ^{ - 1} \left| \psi \right\rangle \\ <br /> \left| \psi \right\rangle = \lambda \Lambda ^{ - 1} \left| \psi \right\rangle \\ <br /> I = \lambda \Lambda ^{ - 1} \\ <br /> \end{array}$
 
Fine except for the last line; you can't drop the state because it's not a generic state; it's an eigenstate of Lambda. You can, however, multiply both sides of the next-to-last line by 1/lambda.
 
Except for the very last line, you're there!
 
Yes, I see. Thanks very much!
 
Hello everyone, I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$ In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”: $$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$ with $$r_{q}(t)=(0,0,z_{q}(t))$$ (I’m aware this isn’t...
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
Back
Top