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This CD defines symbols for basic linear algebra related to eigenvalues.
Regardless of the way of forming vectors and matrices, this CD
deals with eigenvalues, eigenvectors and related concepts.
This symbol represents a Boolean binary function, whose first argument should
be a square matrix A over a ring R and whose second argument should be an
element of the ring R.
Here, the matrix A acts on (row) vectors from the right
and
the scalar lambda is written to the left of the vector v.
When applied to A and lambda, it means that there is an eigenvector vector with
eigenvalue lambda.
Commented Mathematical property (CMP):
If is_eigenvalue(A,lambda), there is a nonzero row vector
v such that v * A = lambda * v.
This symbol represents a Boolean binary function, whose first argument should
be a square matrix A over a ring R and whose second argument should be a
vector v of size rowcount(A) over the ring R.
When applied to A and v, it means that v is a left eigenvector of A.
Commented Mathematical property (CMP):
If is_eigenvector(A,v) then there is a scalar lambda such that
v * A = lambda * v
This symbol represents a binary function, whose arguments should
be a square matrix A and a field F over which the matrix A is defined.
When applied to A and F, it returns a vector whose entries are the eigenvalues
of A contained in F, with multiplicities.
Example:
Consider the matrix A given by
[[0,1],
[1,-1]
].
Its characteristic polynomial is X^2+X-1, and its eigenvalues are
the roots of this equation. Hence, over the rationals, there are no
eigenvalues:
This symbol represents a binary function, whose arguments should
be a square matrix A over a field F and an element lambda of F.
When applied to A and lambda, it returns a matrix whose rows are a basis of the eigenspace
of A with respect to lambda.
