A good place to start is to look at Eugene's Vector article from 1997, *Oh,
No, Not Eigenvalues Again!<http://www.jsoftware.com/jwiki/Doc/Articles/Play141>
* .
The course in college which covered eigenvalues and eigenvectors was not
one of my better courses. I do remember the prof explaining the importance
of eigenvectors as follows: an eigenvector v is one such that (A +/
.*v)=lambda*v . This means that the action of A on v is just a stretching
of v, a particularly simple transformation of v among possible weird and
wonderful transformations, like using kilometers per hour instead of miles
per hour.
If you are serious about computing eigenvalues and eigenvectors I suggest
you use LAPACK. Such computation is tricky involving deep and detailed
knowledge of numerical analysis.
On Fri, Feb 3, 2012 at 3:03 PM, Don Guinn <dongu...@gmail.com> wrote:
>
> 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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