In this section we show you some of Axiom's facilities for computing and eigenvalue manipulating eigenvalues and eigenvectors, also called eigenvector characteristic values and characteristic vectors, characteristic:value respectively. characteristic:vector
Let's first create a matrix with integer entries.
To get a list of the rational eigenvalues, use the operation eigenvalues.
Given an explicit eigenvalue, eigenvector computes the eigenvectors corresponding to it.
The operation eigenvectors returns a list of pairs of values and vectors. When an eigenvalue is rational, Axiom gives you the value explicitly; otherwise, its minimal polynomial is given, (the polynomial of lowest degree with the eigenvalues as roots), together with a parametric representation of the eigenvector using the eigenvalue. This means that if you ask Axiom to solve the minimal polynomial, then you can substitute these roots polynomial:minimal into the parametric form of the corresponding eigenvectors. minimal polynomial
You must be aware that unless an exact eigenvalue has been computed, the eigenvector may be badly in error.
Another possibility is to use the operation radicalEigenvectors tries to compute explicitly the eigenvectors in terms of radicals. radical
Alternatively, Axiom can compute real or complex approximations to the approximation eigenvectors and eigenvalues using the operations realEigenvectors or complexEigenvectors. They each take an additional argument to specify the ``precision'' required. precision In the real case, this means that each approximation will be within of the actual result. In the complex case, this means that each approximation will be within of the actual result in each of the real and imaginary parts.
The precision can be specified as a Float if the results are desired in floating-point notation, or as Fraction Integer if the results are to be expressed using rational (or complex rational) numbers.
If an by matrix has distinct eigenvalues (and therefore eigenvectors) the operation eigenMatrix gives you a matrix of the eigenvectors.
If a symmetric matrix matrix:symmetric has a basis of orthonormal eigenvectors, then basis:orthonormal orthonormalBasis computes a list of these vectors. orthonormal basis