eigen value

can some one tell me the importance of eigen value and eigen vector in
the engineering. I know that it gives input impedance when eigen
vector is input and it also gives the frequency of an oscillator, but
how? what more applications does it have?

pankajjd@rediffmail.com wrote:

> can some one tell me the importance of eigen value and eigen vector in
> the engineering. I know that it gives input impedance when eigen
> vector is input and it also gives the frequency of an oscillator, but
> how? what more applications does it have?

OK, so the last time I heard of eigen value/vector was > 30 yrs ago.
[ IIRC they were defined as Matrix * eigenvector = eigenvalue * 
eigenvector ]

I don't see a connection to either "input impedance" or "frequency of 
an oscillator".

What am I missing?

Richard Owlett wrote:

(snip regarding eigenvalues and eigenvectors)

> OK, so the last time I heard of eigen value/vector was > 30 yrs ago.
> [ IIRC they were defined as Matrix * eigenvector = eigenvalue * 
> eigenvector ]

> I don't see a connection to either "input impedance" or "frequency of an 
> oscillator".

They come from the normal modes of oscillating systems, as one
example.  The more obvious modes may be linear combinations of
modes with different frequencies, and so won't have a true
frequency, that is, they won't be periodic.

A favorite demonstration is a spring and mass system with both
a stretch mode and torsion (rotational) mode.  If the two normal
modes, which are linear combinations of the stretch and tortional
modes, have nearly the same frequency, an interesting effect is
observed.  If you start the system in a stretch mode it will slowly
convert to the tortional mode, and then slowly convert back again.

Many other systems have such oscillatory behavior when excited in
a mode that is not the normal mode of the system.

(One of the more recent discoveries is that neutrinos from the
sun are not a pure state (normal mode), and oscillate between
electron neutrinos and other types of neutrinos such as mu
neutrinos and tau neutrinos.)

-- glen

Richard Owlett <rowlett@atlascomm.net> wrote in message news:<101qn1re2jvo965@corp.supernews.com>...
> pankajjd@rediffmail.com wrote:
> 
> > can some one tell me the importance of eigen value and eigen vector in
> > the engineering. I know that it gives input impedance when eigen
> > vector is input and it also gives the frequency of an oscillator, but
> > how? what more applications does it have?
> 
> OK, so the last time I heard of eigen value/vector was > 30 yrs ago.
> [ IIRC they were defined as Matrix * eigenvector = eigenvalue * 
> eigenvector ]
> 
> I don't see a connection to either "input impedance" or "frequency of 
> an oscillator".
> 
> What am I missing?

The connection is there. See, for instance, the differential equation 

  df(t)/dt = f(t)                                                  [1]

With a little twist of the imagination, you can say the d/dt part 
is an "operator" (i.e. a generalized matrix) and call it L. 
Mathematicians throw in the eigenvalue lambda = 1 for good measure 
such that [1] can be written as

   Lf = lambda f.                                                  [2]

The solution f to the equation [1] will behave just as the eigenvector 
in matrixes. Most (all?) linear differential equations that are of 
interest in mathematical physics behave like [2], although with somewhat 
more complicated L operators. In maths, this is what "Sturm-Liouville 
theory" is all about. Real Analysis is a very convenient tool for 
working with such problems.

Rune

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0402020020.69bc34d9@posting.google.com>...
> Richard Owlett <rowlett@atlascomm.net> wrote in message news:<101qn1re2jvo965@corp.supernews.com>...
> > pankajjd@rediffmail.com wrote:
> > 
> > > can some one tell me the importance of eigen value and eigen vector in
> > > the engineering. I know that it gives input impedance when eigen
> > > vector is input and it also gives the frequency of an oscillator, but
> > > how? what more applications does it have?
> > 
> > OK, so the last time I heard of eigen value/vector was > 30 yrs ago.
> > [ IIRC they were defined as Matrix * eigenvector = eigenvalue * 
> > eigenvector ]
> > 
> > I don't see a connection to either "input impedance" or "frequency of 
> > an oscillator".
> > 
> > What am I missing?
> 
> The connection is there. See, for instance, the differential equation 
> 
>   df(t)/dt = f(t)                                                  [1]
> 
> With a little twist of the imagination, you can say the d/dt part 
> is an "operator" (i.e. a generalized matrix) and call it L. 
> Mathematicians throw in the eigenvalue lambda = 1 for good measure 
> such that [1] can be written as
> 
>    Lf = lambda f.                                                  [2]
> 
> The solution f to the equation [1] will behave just as the eigenvector 
> in matrixes. Most (all?) linear differential equations that are of 
> interest in mathematical physics behave like [2], although with somewhat 
> more complicated L operators. In maths, this is what "Sturm-Liouville 
> theory" is all about. Real Analysis is a very convenient tool for 
> working with such problems.
> 
> Rune

I think Rune is absolutely  right. Eigen vector/values are used all
over quantam mechanics/mathemical physics. As far as engineering is
concerned it is also used in communication system- mainly in linear
algebra(matrix computation,least square solutions). I guess if you
open IEEE trans. on comm. you may notice many papers  using linear
algebra(mathematical tool). Although
there are various solutions but eigen vector/values method can  be one
of the solution provider tools. For details, can see "Matrix
computation" by Golub,Loan.

Santosh

Hello Santosh and others,
One simple case of eigenvalues/eigenfunctions we all use in DSP concerns the
applications of linear filters. We all know when you put a sinusoid into a
linear filter you get the same sinusoid out albeit scaled in amplitude and
shifted in phase. For this system the sinusoid is an eigenfunction and the
complex number representing its amplitude and phase shift is the
corresponding eigenvalue. A list of the applications of eigensystems could
fill a book or two. I just thought I'd mention one that we have all used
without thinking about its being part of an eigensystem - but it is. In fact
a consequence of this is linear systems don't produce frequencies that
weren't present in the first place.

-- 
Clay S. Turner, V.P.
Wireless Systems Engineering, Inc.
Satellite Beach, Florida 32937
(321) 777-7889
www.wse.biz
csturner@wse.biz



"santosh nath" <santosh.nath@ntlworld.com> wrote in message
news:6afd943a.0402020852.3fe82343@posting.google.com...
> allnor@tele.ntnu.no (Rune Allnor) wrote in message
news:<f56893ae.0402020020.69bc34d9@posting.google.com>...
> > Richard Owlett <rowlett@atlascomm.net> wrote in message
news:<101qn1re2jvo965@corp.supernews.com>...
> > > pankajjd@rediffmail.com wrote:
> > >
> > > > can some one tell me the importance of eigen value and eigen vector
in
> > > > the engineering. I know that it gives input impedance when eigen
> > > > vector is input and it also gives the frequency of an oscillator,
but
> > > > how? what more applications does it have?
> > >
> > > OK, so the last time I heard of eigen value/vector was > 30 yrs ago.
> > > [ IIRC they were defined as Matrix * eigenvector = eigenvalue *
> > > eigenvector ]
> > >
> > > I don't see a connection to either "input impedance" or "frequency of
> > > an oscillator".
> > >
> > > What am I missing?
> >
> > The connection is there. See, for instance, the differential equation
> >
> >   df(t)/dt = f(t)                                                  [1]
> >
> > With a little twist of the imagination, you can say the d/dt part
> > is an "operator" (i.e. a generalized matrix) and call it L.
> > Mathematicians throw in the eigenvalue lambda = 1 for good measure
> > such that [1] can be written as
> >
> >    Lf = lambda f.                                                  [2]
> >
> > The solution f to the equation [1] will behave just as the eigenvector
> > in matrixes. Most (all?) linear differential equations that are of
> > interest in mathematical physics behave like [2], although with somewhat
> > more complicated L operators. In maths, this is what "Sturm-Liouville
> > theory" is all about. Real Analysis is a very convenient tool for
> > working with such problems.
> >
> > Rune
>
> I think Rune is absolutely  right. Eigen vector/values are used all
> over quantam mechanics/mathemical physics. As far as engineering is
> concerned it is also used in communication system- mainly in linear
> algebra(matrix computation,least square solutions). I guess if you
> open IEEE trans. on comm. you may notice many papers  using linear
> algebra(mathematical tool). Although
> there are various solutions but eigen vector/values method can  be one
> of the solution provider tools. For details, can see "Matrix
> computation" by Golub,Loan.
>
> Santosh

Rune Allnor wrote:
> Richard Owlett <rowlett@atlascomm.net> wrote in message news:<101qn1re2jvo965@corp.supernews.com>...
> 
>>pankajjd@rediffmail.com wrote:
>>
>>
>>>can some one tell me the importance of eigen value and eigen vector in
>>>the engineering. I know that it gives input impedance when eigen
>>>vector is input and it also gives the frequency of an oscillator, but
>>>how? what more applications does it have?
>>
>>OK, so the last time I heard of eigen value/vector was > 30 yrs ago.
>>[ IIRC they were defined as Matrix * eigenvector = eigenvalue * 
>>eigenvector ]
>>
>>I don't see a connection to either "input impedance" or "frequency of 
>>an oscillator".
>>
>>What am I missing?
> 
> 
> The connection is there. See, for instance, the differential equation 
> 
>   df(t)/dt = f(t)                                                  [1]
> 
> With a little twist of the imagination, you can say the d/dt part 
> is an "operator" (i.e. a generalized matrix) and call it L. 
> Mathematicians throw in the eigenvalue lambda = 1 for good measure 
> such that [1] can be written as
> 
>    Lf = lambda f.                                                  [2]
> 
> The solution f to the equation [1] will behave just as the eigenvector 
> in matrixes. Most (all?) linear differential equations that are of 
> interest in mathematical physics behave like [2], although with somewhat 
> more complicated L operators. In maths, this is what "Sturm-Liouville 
> theory" is all about. Real Analysis is a very convenient tool for 
> working with such problems.
> 
> Rune

As soon as you said 'the d/dt part is an "operator" ' ...
[nightmare deleted ;] One prof didn't survive that course either ( he 
was a PURE algebraist trying to teach class of 50% engineers -- got 
shipped off to a think tank ;)!

Richard Owlett wrote:

   ...

> As soon as you said 'the d/dt part is an "operator" ' ...
> [nightmare deleted ;] 

   ...

It's a matter of language again. Suppose we adopt Heavyside's notation D 
  instead of d/dt; does that help? If not, try derivative_of(). It's all 
the same meaning. It was Heavyside who introduced "operator" in this 
context, calling his way of solving equations "operational calculus".

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
> Richard Owlett wrote:
> 
>   ...
> 
>> As soon as you said 'the d/dt part is an "operator" ' ...
>> [nightmare deleted ;] 
> 
> 
>   ...
> 
> It's a matter of language again. Suppose we adopt Heavyside's notation D 
>  instead of d/dt; does that help? If not, try derivative_of(). It's all 
> the same meaning. It was Heavyside who introduced "operator" in this 
> context, calling his way of solving equations "operational calculus".
> 
> Jerry

I think you misinterpreted my response.
It was more on order of "Oh no not *THAT* nn inch thick set of 
prepared class notes.

> can some one tell me the importance of eigen value and eigen vector in
> the engineering. 

An interesting application is found in speech enhancement: if you
think of speech segment as a vector of length n, then you can expand
it into its projections onto n or more other vectors (signals). The
Fourier transform is one posibility where the vectors are orthogonal
exponential functions. In the matrix notation the signal can be
represented as product of the expansion vector (spectrum) and the
fixed basis matrix (exponential functions).
In a similar manner you can represent signal (vector) as product of
its eigen-values and its eigen-vector matrix. This expansion is in so
far specific as all eigen-vectors are orthogonal to the signal vector.
But the interesting thing is that this also implies that the
eigenvectors follows the single signal patterns that make the
composite signal. It turns out that in the speech + noise signal
speech and noise are projected onto separate vectors. If you can guess
which of the eigen-vectors belongs to noise you can kill it by setting
the appropriate eigen-value to zero.

The expansion above is called Karhunen-Loeve Transform or KLT (but it
is not trransform).

Tarik