5 replies to this topic

[EigenCake]

Hi, all!

On my Classpad 330, I tried to evaluate "eigVl([[a,b][c,d]])", where a, b, c, d are unknown constants. It returned "Wrong argument type". Please note that if evaluate "eigVl([[1,2][3,4]])", it works.

I also tried "eigVc([[a,b][c,d]])". It appeared the same thing. So how can I find eigenvalue and eigenvector please?

Thanks very much!

[EigenCake]

By the way, does anyone know whether TI-Nspire series is able to evaluate this or not? If yes, I'll immediately throw Classpad 330 into garbage can and buy an Nspire.

[supergems]

With HP 50g in RPN:

[[ 'A' 'B' ] [ 'C' 'D' ]] EGVL --> [ '(A+D-\v/(A^2-2*D*A+(4*C*B+D^2)))/2' '(A+D+\v/(A^2-2*D*A+(4*C*B+D^2)))/2' ]

[[ 'A' 'B' ] [ 'C' 'D' ]] EGV --> [[ 'B' 'B' ] [ '-((A-D+\v/(A^2-2*D*A+(4*C*B+D^2)))/2)' '(-(A-D)+\v/(A^2-2*D*A+(4*C*B+D^2)))/2' ]] [ '(A+D-\v/(A^2-2*D*A+(4*C*B+D^2)))/2' '(A+D+\v/(A^2-2*D*A+(4*C*B+D^2)))/2' ]

EGVL computes the eigenvalues of a square matrix.
EGV computes the eigenvalues and right eigenvectors for a square matrix.

[pan.gejt]

TI-Nspire CAS and TI-89 Titanium are not able to evaluate this operation also.
Error message: Invalid data type

[MicroPro]

Hello EigenCake and welcome to the UCF!

Your problem is actually related to mathematics. I never studied eigenvalues well, but as my math knowledge tells me eigenvalues for a particular matrix are not always unique (neither are eigenvectors). ClassPad may use numeric methods or special matrix reductions to find the eigenvalues/vectors of matrices. This means it can't recognize algebraic expressions and variables in its calculations. And therefore can't give you a single algebraic expression that can represent the formula of the eigenvalue of every matrix.

It's like finding roots of complicated equations. For example, you can use a numeric method (bisection, newton, etc) to solve sin(x^2)+x^2+x^4-0.5=0, but there doesn't exist an algebraic formula that solves every equation of this kind.

Also see this thread about eigenvectors in ClassPad: http://www.casiocalc...?showtopic=3611

With HP 50g in RPN:
[[ 'A' 'B' ] [ 'C' 'D' ]] EGVL --> [ '(A+D-\v/(A^2-2*D*A+(4*C*B+D^2)))/2' '(A+D+\v/(A^2-2*D*A+(4*C*B+D^2)))/2' ]
[[ 'A' 'B' ] [ 'C' 'D' ]] EGV --> [[ 'B' 'B' ] [ '-((A-D+\v/(A^2-2*D*A+(4*C*B+D^2)))/2)' '(-(A-D)+\v/(A^2-2*D*A+(4*C*B+D^2)))/2' ]] [ '(A+D-\v/(A^2-2*D*A+(4*C*B+D^2)))/2' '(A+D+\v/(A^2-2*D*A+(4*C*B+D^2)))/2' ]

This shows hp's CAS uses special matrix reductions and it's strong enough to do all its internal calculations with CAS enabled.

[EigenCake]

Ok, thank you very much for the reponses from these kind-hearted people!

To Micropro:
Well, I can not remember that much as well, and I have difficulty to find my textbook, however, the following stuff is something first coming into my mind.

The eigenvectors are the elements from eigenspace, which is spanned by the basis vectors and zero vector. Those basis vectors are special eigenvectors, because they are always linearly independent - in other loose words - they are ALWAYS "unique". Any eigenvector from the eigenspace can be expressed by the linear combination of the bases. However, the basis vectors may not be orthogonal. In order to have orthogonal bases, Gram-Schmidt orthogonalization may be needed.

The number of distinct eignvalues are equal to or less than the dimension of matrix. If all eigenvalues from one particular matrix are distinct and equal to the dimension of matrix, then the eigenvector associated to each eigenvalue is linearly independent from any other eigenvector associated to another eigenvalue; If the two eigenvalues are repeated, then there will be two linearly independent eigenvectors associated to a single eigenvalue, and there will be no linearly independent eigenvector associated to the other eigenvalue, in other words, they share the eigenvectors.


To supergems,
Perhaps, I should buy a HP 50g RPN. One thing is apparent that the Computer Algerbra System in Classpad is poor!


To pan.gejt
Thanks! That comeforts me a little bit; otherwise, my Classpad 330 will be murdered, then shattered and shredded by my evil hands.


Ok, let's go back to our original question:

By using Maple,
> Matrix(2, 2, {(1, 1) = a, (1, 2) = b, (2, 1) = c, (2, 2) = d});

I'll get two eigenvectors as shown below:
Vector(2, {(1) = (1/2)*d+(1/2)*a+(1/2)*sqrt(d^2-2*d*a+a^2+4*c* B) , (2) = (1/2)*d+(1/2)*a-(1/2)*sqrt(d^2-2*d*a+a^2+4*c* B) });

And the Characteristic polynomial:
lambda^2+(-d-a)*lambda-c*b+d*a=0;

In order to use Classpad 330 to find the eigenvalue, then I have to find the Characteristic polynomial by hand (luckily, it is not tedious!) , then in Classpad 330 and type:
solve(x^2+(-d-a)*x-c*b+d*a=0,x)

This will yield the same result as Maple does.

To evaluate the eigenvectors, Maple yields two eigenvectors:
Matrix(2, 2, {(1, 1) = b/((1/2)*d-(1/2)*a+(1/2)*sqrt(d^2-2*d*a+a^2+4*c* B) ), (1, 2) = b/((1/2)*d-(1/2)*a-(1/2)*sqrt(d^2-2*d*a+a^2+4*c* B) ), (2, 1) = 1, (2, 2) = 1});

whereas Classpad 330 is unable to do so, even I tried to use:
ref([[a-x, b][c, d-x]])
which does not yield the wanted the row-echelon-form.

Poor Classpad~

Edited by EigenCake, 19 August 2011 - 05:00 AM.