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Applications of Eigenvectors!?

Discussion in 'General Electronics' started by Steven O., Dec 17, 2003.

  1. Steven O.

    Steven O. Guest

    Math is a hobby for me. I've been reading up on Eigenvectors and
    Eigenvalues. It get the manipulations involved, but can't imagine the
    applications -- and the books I have don't help. Can people provide a
    few examples?

    Specific examples, if possible -- not just, they are used in
    electronics, or physics, or whatever, but rather, something like:

    M is the matrix which describes such-and-such physical property or
    transformation or process, its eigenvectors V correspond to such and
    such property, and the eigenvalues of V and M indicate such-and-such.

    Thanks in advance for all replies.
    Steve O.

    Standard Antiflame Disclaimer: Please don't flame me. I may actually *be* an idiot, but even idiots have feelings.
     
  2. [email protected]_ALL_Zs_AND_ALL_BETWEEN_ZZComm.com wrote on 12/16/03 10:27
    PM:
    One example is to describe states of polarized light. There are two
    eigenstates, say horizontally and vertically polaraized light. Any
    polarization state is represented as a sum of these two states multiplied by
    complex constants. That includes circular and elliptical polarization.
    Operations on these eigenvectors represent what happens when the light goes
    through a waveplate.

    Electron spins can also b represented by the same mathematics.

    An amusing series authored by Joseph Slepian some decades ago, probably in
    the 30s, maybe 40s appearing in the Transactions of the IEEE. You will need
    a good technical library to dig that out. It is about someone financed to a
    technical education by an uncle. The lad reduced the inventory of his
    uncle's custom fruit salad business by representing mixtures of fruit as
    vectors. The method depended upon making eigenvector combinations of fruit.

    Bill
     
  3. A common use of eigenvalues and eigenvectors is in the analysis of dynamic
    mechanical systems.
    Given an undamped mechanical system described by the differential equations
    [M](d^2x)/(dt^2)+[K]u=0
    where M is mass, K is spring stiffness
    (d^2x)/(dt^2) is acceleration and x is position,
    the eigenvalues of the system notes the squared ressonant frequencies of the
    system and the eigenvectors are the decomposed patterns of motion.

    --
    ----------------------------
    Christopher Grinde
    Ph.D student
    Mobile:+47 91137588
    Tlph: +47 33037717
    Web:http://cg.ans.hive.no
     
  4. Imagine P being the matrix of transition probabilities from
    one state of a system to another of some system.
    P_ij is the probability that the system goes from state i to
    state j. The sum of each row is one:
    sum( P_ij, j = 1...n ) = 1.
    This is the transition matrix of a so-called Markov chain.

    Under certain circumstances the infinite matrix product
    limit converges such that
    limit( P_ij^(n), n-->infinity) = p_j for all i,j.
    where [ p_j, j=1...n ] is the limit vector of the probabilities
    of the system being in the different states.
    Here P_ij^(n) is the i,j element of the product matrix P^n,
    with the transition probabilities from state i to state j after
    n steps (as opposed to after 1 step as P_ij).

    In stead of calculating the limit, one can try to find the
    vector [ p_i ] of the probabilities of the initial states,
    such that these probabilities are not influenced by the
    evolution of the system, i.o.w. find the vector [ p_i ]
    such that
    sum( p_i * P_ij, i=1...n ) = p_j for all j,
    i.o.w. find an eigenvector with eigenvalue 1 of the
    transposed matrix P^t.
    This eigenvector with probabilities of the initial system
    being in the different states, does not change when the
    sytem evolves.

    Dirk Vdm
     
  5. dtn

    dtn Guest

    If M is a matrix of force constants of a molecule, the eigenvalues are the
    vibrational frequencies and the eigenvectors are the normal modes.
    Ref. I. Wilson, ca. 1930's.

    *be* an idiot, but even idiots have feelings.
     
  6. The eigenvalues are the square of the vibrational frequencies. You get
    negative eigenvalues at a transition stucture or any stationary point
    other than a local minimum.
     
  7. Daniel Grubb

    Daniel Grubb Guest

    1) If M is the inertial matrix, the eigenvectors are those angular
    velocities where the angular momentum is parallel to the angular velocity.

    2) If M is the 'Hamiltonian' matrix, the eigenvalues are the allowed
    energies of the system. The eigenvectors represent the 'stable' states
    of the system.

    3) (generalization of 2)) If M is the matrix that describes an observble,
    the eigenvalues are the allowed measured values of that observable.

    4) In analysis of small oscillations, there are matrices M and V representing
    the masses and potentials. The solutions to the eigenvalue problem
    det(-Mw^2 +V)=0 give the frequencies w of the system. The eigenvectors
    correspond to normal modes of the system.

    5) If M represents a rotation matrix, the eigenvector represents the axis
    of rotation. (Unless the rotation is through an angle of 0 there is just
    one real eigenvalue.)

    That's a start :)

    --Dan Grubb
     
  8. Lee Rudolph

    Lee Rudolph Guest

    You might want to revise this one a bit.

    Lee Rudolph
     
  9. Steven O. ([email protected]_ALL_Zs_AND_ALL_BETWEEN_ZZComm.com) wrote:
    : Math is a hobby for me. I've been reading up on Eigenvectors and
    : Eigenvalues. It get the manipulations involved, but can't imagine the
    : applications -- and the books I have don't help. Can people provide a
    : few examples?

    : Specific examples, if possible -- not just, they are used in
    : electronics, or physics, or whatever, but rather, something like:

    : M is the matrix which describes such-and-such physical property or
    : transformation or process, its eigenvectors V correspond to such and
    : such property, and the eigenvalues of V and M indicate such-and-such.

    All quantum mechanics of bound states involve finding the eigenvectors of
    Hermetian matrices. Since you seem to have a grasp of matrix math, try
    reading Levine's textbook on quantum chemistry. It's probably the best
    introduction to QM. Maybe Atkin's undergrad Physical Chemistry book would
    do as well, but it's also full of thermodynamics and kinetics (maybe not a
    bad idea to learn those as well).

    If you have a few thousand $US to spend, you can get Gaussian for PC, and
    solve some of these massive problems yourself (you'll get to do the inputs
    and see the outcomes of real quantum-chemical problems). You'll need a
    state-of-the-art, high-end PC.

    Hope I helped.


    --
    --
    William "Dave" Thweatt
    Robert E. Welch Postdoctoral Fellow
    Chemistry Department
    Rice University
    Houston, TX

     
  10. Guest

    Oops, quite right.

    --Dan Grubb
     
  11. Lee Rudolph

    Lee Rudolph Guest

    And there, class, is a nice example of a participant in an
    argument^Wdiscussion gracefully doing a 180.

    Lee Rudolph
     
  12. eigenvalue -1

    Dirk Vdm
     
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