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Applications of Eigenvectors!?

S

Steven O.

Jan 1, 1970
0
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.
 
R

Repeating Rifle

Jan 1, 1970
0
[email protected]_ALL_Zs_AND_ALL_BETWEEN_ZZComm.com wrote on 12/16/03 10:27
PM:
Specific examples, if possible -- not just, they are used in
electronics, or physics, or whatever, but rather, something like:

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
 
C

Christopher Grinde

Jan 1, 1970
0
Steven said:
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.

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
 
D

Dirk Van de moortel

Jan 1, 1970
0
Steven O. said:
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.

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
 
D

dtn

Jan 1, 1970
0
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.

Steven O. said:
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.
 
B

Brian Salter-Duke

Jan 1, 1970
0
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.

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.
 
D

Daniel Grubb

Jan 1, 1970
0
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.

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
 
L

Lee Rudolph

Jan 1, 1970
0
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.)

You might want to revise this one a bit.

Lee Rudolph
 
W

William David Thweatt

Jan 1, 1970
0
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
[email protected]
[email protected]
 
L

Lee Rudolph

Jan 1, 1970
0
Oops, quite right.

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

Lee Rudolph
 
D

Dirk Van de moortel

Jan 1, 1970
0
Lee Rudolph said:
And there, class, is a nice example of a participant in an
argument^Wdiscussion gracefully doing a 180.

eigenvalue -1

Dirk Vdm
 
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