For an introduction wikipedia is the place to go.
http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
- Bo
>________________________________
> Fra: Don Guinn <dongu...@gmail.com>
>Til: Programming forum <programming@jsoftware.com>
>Sendt: 0:03 lørdag den 4. februar 2012
>Emne: [Jprogramming] Eigen or Eigan?
>
>This is a long one that may be turned into a J essay if I ever figure it
>all out. I will not feel bad if you skip reading it.
>
>A couple of weeks ago I dug up a topic I studied only enough to get through
>a test in college. Eigenvalues. First, how is it spelled? Both Eigenvalue
>and Eiganvalue work on GOOGLE. Second, lots of pages described how to find
>Eigenvalues, but nobody tackled Eigenvectors. And then, what are
>Eigenvectors good for? Lots of pages talk about applications but I thought
>it better to understand Eigenvalues and Eigenvectors before I tackled
>applications.
>
>I am not through figuring out Eigenvalues yet, but I am tossing out what I
>have so far as Linda is looking for topics for the classroom and a couple
>of things here she might find interesting, if she has not already done
>them. Second, I would appreciate some suggestions as to where to go next.
>
>I searched the web on Eigenvalues and I looked at the J Essay "Eigenvalues"
>by John Randall and I could use them to calculate Eigenvalues quite nicely.
>Nothing about finding Eigenvectors anywhere. Rather than getting involved
>in the issues of how to best calculate Eigenvalues I wanted to start from
>scratch. I wanted to use the equation (Ax-λIx)=0 for finding Eigenvalues as
>nearly as I could in J. There are references in J about polynomial
>arithmetic. Given that and the ability of J adverbs and conjunctions to
>take defined verbs as arguments, can this equation be translated into J?
>
>I chose to represent polynomials similarly to their description in the verb
>"p." only box the coefficients. For example:
> (x^2 + 3x + 2) <==> <2 3 1
>
>Defined a few tools for doing polynomial arithmetic:
> Ptimes=:+//.@:(*/)&.>
> Pminus=:(-/@:,:)&.>
> Pdet=:Pminus/ . Ptimes
> Pconstant=:(<@,"0`]@.(*@L.))
>
>Linda - Every math senior has had to do polynomial arithmetic and if they
>are anything like I was, they found it easy to understand but tedious and
>boring to do on paper. J is unique as a programming language that can do
>polynomial arithmetic easily. Also, after having to calculate the
>determinants of 4 by 4 matrices using pencil and paper had my teacher been
>able to show me a machine that would easily calculate determinants and do
>polynomial arithmetic, he would surly have had my attention.
>
>(I am sending this in as Courier New and hopefully will display OK)
>
>Books give three steps to determine the Eigenvalues of a matrix.
>1. Build a polynomial matrix of the matrix with λ subtracted from the main
>diagonal.
>Given matrix A:
> ]A=:Pconstant 1 2 1,6 _1 0,:_1 _2 _1
>+--+--+--+
>|1 |2 |1 |
>+--+--+--+
>|6 |_1|0 |
>+--+--+--+
>|_1|_2|_1|
>+--+--+--+
> Lamda=:<0 1 NB. Lamda is the polynomial "0 + x"
> ]AL=:A Pminus (Lamda Ptimes =i.3)
>+----+-----+-----+
>|1 _1|2 0 |1 0 |
>+----+-----+-----+
>|6 0 |_1 _1|0 0 |
>+----+-----+-----+
>|_1 0|_2 0 |_1 _1|
>+----+-----+-----+
>
>2. Calculate the Characteristic Polynomial
> ]CP=:Pdet AL
>+----------+
>|0 12 _1 _1|
>+----------+
>
>3. Find its factors. (Nice that J has a tool to factor polynomials)
> ;}.p.>CP
>_4 3 0
>
>This is just how my textbook described calculating Eigenvalues, only here
>it's done in J. I'm sure that there are faster and more accurate methods
>for finding Eigenvalues; however, it was fun to so directly replace
>traditional mathematical notation used in the book with J. And still have
>it read like the book.
>
>My next challenge was finding the Eigenvectors. Not too hard to do with
>pencil and paper, but getting J to do it is another story. One can easily
>build a matrix to find an Eigenvector from each Eigenvalue by putting in
>values for A-Iλ; however, a given Eigenvalue may have more than one
>Eigenvector associated with it. For a 3 by 3 matrix it is essentially 3
>equations with 3 unknowns with no constant and all equal to zero. And all
>unknowns = zero is not acceptable. The matrix is singular as λ is a root so
>the rows are not independent. A row can be eliminated and arbitrarily pick
>a value for one of the unknowns then solve for the other two.
>
>This is where I am not sure as to what to do. Picking n-1 adjacent rows
>works well when all Eigenvalues are unique, but when an Eigenvalue is
>repeated I am not sure if adjacent rows is sufficient for finding
>solutions. I am not sure about the code below for finding Eigenvectors when
>multiple orthogonal Eigenvectors satisfy an Eigenvector, but it hasn't
>failed yet.
>
>The expression (M -: EV dot (E*=i.#E) dot %. EV) is supposed to hold,
> where M is the matrix,
> E is the list of Eigenvalues,
> and EV is the matrix of corresponding Eigenvectors in columns
>
>This test is performed by Eigen_test to verify that the calculation is
>correct.
>
>NB. Start of script __________________________
>
>NB. Polynomial arithmetic operations:
>Ptimes=:+//.@:(*/)&.>
>Pminus=:(-/@:,:)&.>
>Pdet=:Pminus/ . Ptimes
>Pconstant=:(<@,"0`]@.(*@L.))
>
>NB. Eigen names
>Lamda=:<0 1
>Characteristic_Polynomial=:13 : 'Pdet y Pminus Lamda Ptimes =i.#y'
>Eigenvalues=:13 : ';}.p.>Characteristic_Polynomial y'
>
>Eigenvectors=:4 : 0
>inv=.%. :: 0:
>for_i. ((#y)-x)<;._3 y do.
> 'v c'=.(-x)(([:|:}.);{.)|:;i
> if. $im=.inv v do.
> <(-=i.x),~|:c+/ .*|:im
> return.
> end.
>end.
>('No Eigenvector found for Eigenvalue ',":x) (13!:8) 3
>)
>
>Eigen_test=:4 : 0
>dot=.+/ .*
>'e v'=.y
>if. -.x-:v dot (e*=i.#e) dot %. v do.
>('Eigenvalues and Eigenvectors invalid',,LF,.":y) (13!:8) 3
>end.
>y
>)
>
>Eigens=:3 : 0
>py=.Pconstant y NB. Convert y to polynomial form if necessary.
>'vals counts'=.(](];[:+/=/)~.)Eigenvalues py
>m=.y-"2 vals*"0 2=i.#y
>vects=.,.&.>/counts Eigenvectors"0 2 m
>y Eigen_test r=.(counts#vals);vects
>)
>
>NB. End of Script _____________________________
>
>The verb Eigens puts the calculations together and Eigenvalues and
>Eigenvectors are returned as two boxed items.
>
> ]A=:1 2 1,6 _1 0,:_1 _2 _1
>1 2 1
>6 _1 0
>_1 _2 _1
> Eigens A
>+------+-----------+
>|_4 3 0| 1 1 1r13|
>| |_2 3r2 6r13|
>| |_1 _1 _1|
>+------+-----------+
>
>This is the same matrix covered earlier. Where did the rationals come from?
>I never used rationals.
>
> ]B=:3 2 4,2 0 2,:4 2 3
>3 2 4
>2 0 2
>4 2 3
> Eigens B
>+-------+-----------+
>|8 _1 _1| _1 1r2 1|
>| |_1r2 _1 0|
>| | _1 0 _1|
>+-------+-----------+
>
>B has the Eigenvalue _1 occur twice. Therefore it has two Eigenvectors
>associated with it. It is extremely difficult to compare these Eigenvectors
>with those calculated other ways as linear combinations of them can look
>quite different.
>
> Eigens 7#,:>:i.7
>+--------------+--------------------+
>|28 0 0 0 0 0 0|_1 2 3 4 5 6 7|
>| |_1 _1 0 0 0 0 0|
>| |_1 0 _1 0 0 0 0|
>| |_1 0 0 _1 0 0 0|
>| |_1 0 0 0 _1 0 0|
>| |_1 0 0 0 0 _1 0|
>| |_1 0 0 0 0 0 _1|
>+--------------+--------------------+
>
>Interesting.
>
>A matrix can be raised to a power given its Eigenvalues and Eigenvectors by
>applying the same formula as used in Eigen_test to verify the solution.
>Back to A again.
> ]'e v'=.Eigens A
>+------+-----------+
>|_4 3 0| 1 1 1r13|
>| |_2 3r2 6r13|
>| |_1 _1 _1|
>+------+-----------+
> dot=:+/ .*
> v dot ((*:e)*=i.3) dot %. v
>12 _2 0
> 0 13 6
>_12 2 0
> A dot A
>12 _2 0
> 0 13 6
>_12 2 0
>
>Raise e, the Eigenvalues, to other powers and this still works.
>
>For the fun of it I tried square root.
> ]rA=:v dot ((%:e)*=i.3) dot %. v
> 1.31966j0.642857 0.494872j_0.571429 0.329914j_0.214286
> 1.97949j_1.28571 0.742307j1.14286 0.494872j0.428571
>_1.31966j_0.642857 _0.494872j0.571429 _0.329914j0.214286
> rA dot rA
>1j_5.55112e_17 2j5.55112e_17 1j2.77556e_17
> 6 _1 _1.66533e_16j2.77556e_17
>_1j5.55112e_17 _2j_5.55112e_17 _1j_2.77556e_17
> A-rA dot rA
> 8.88178e_16j5.55112e_17 4.44089e_16j_5.55112e_17 2.22045e_16j_2.77556e_17
> 0 2.22045e_16 1.66533e_16j_2.77556e_17
>_8.88178e_16j_5.55112e_17 _4.44089e_16j5.55112e_17 _2.22045e_16j2.77556e_17
>
>OK, got some round-off errors but pretty close. I don't know if this is
>supposed to work or not.
>
>
>I am not at all comfortable with the verb Eigenvectors, but it has seems to
>have worked OK so far.
>
>I would appreciate comments and suggestions and where to go to better
>understand the applications of Eigenvalues and Eigenvectors.
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>
>
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