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From Lojban to English :
Word: aigne [jbovlaste]
Type: fu'ivla
Gloss Word: eigen- in the sense of "prefix; mathematical/physical"
Gloss Word: eigenspace-generalization exponent in the sense of "mathematical"
Gloss Word: eigenspace (generalized) in the sense of "mathematical; linear transformation, vector space; generalized according to equation aforementioned"
Gloss Word: eigenvalue in the sense of "mathematical; of a square matrix/linear transformation"
Gloss Word: eigenvector in the sense of "mathematical; linear transformation/square matrix"
Gloss Word: multiplicity (algebraic) of eigenvalue in the sense of "mathematical; degree of linear terms in characteristic polynomial of the linear transformation/square matrix; useful for Jordan canonical form computations; algebraic mulitplicity"
Gloss Word: self-preserving vector under mapping/transformation in the sense of "mathematical; (perfect preservation not implied: dilation/contraction by scalar, including by scalar zero (0), allowed)"
Definition: x1 is an eigenvalue (or zero) of linear transformation/square
matrix x2, associated with/'owning' all vectors in generalized
eigenspace x3 (implies neither nondegeneracy nor degeneracy;
default includes the zero vector) with
'eigenspace-generalization' power/exponent x4 (typically and
probably by cultural default will be 1), with algebraic
multiplicity (of eigenvalue) x5
Notes: For any eigenvector v in generalized eigenspace x3 of linear
transformation x2 for eigenvalue x1, where I is the identity
matrix/transformation that works/makes sense in the context,
the following equation is satisfied: $((x_2 - x_1 I)^x_4)v =
0$. When the argument of x4 is 1, the generalized eigenspace x3
is simply a strict/simple/basic eigenspace; this is the typical
(and probable cultural default) meaning of this word. x4 will
typically be restricted to integer values k > 0. x2 should
always be specified (at least implicitly by context), for an
eigenvalue does not mean much without the linear transformation
being known. However, since one usually knows the said linear
transformation, and since the basic underlying relationship of
this word is "eigen-ness", the eigenvalue is given the primary
terbri (x1). When filling x3 and/or x4, x2 and x1 (in that
order of importance) should already be (at least contextually
implicitly) specified. x3 is the set of all eigenvectors of
linear transformation x2, endowed with all of the typical
operations of the vector space at hand. The default includes
the zero vector (else the x3 eigenspace is not actually a
vector space); normally in the context of mathematics, the zero
vector is not considered to be an eigenvector, but by this
definition it is included. Thus, a Lojban mathematician would
consider the zero vector to be an (automatic) eigenvector of
the given (in fact, any) linear transformation (particularly
ones represented by a square matrix in a given basis). This is
the logically most basic definition, but is contrary to typical
mathematical culture. This word implies neither nondegeneracy
nor degeneracy of eigenspace x3. In other words there may or
may not be more than one linearly independent vector in the
eigenspace x3 for a given eigenvalue x1 of linear
transformation x2. x3 is the unique generalized eigenspace of
x2 for given values of x1 and x4. x1 is not necessarily the
unique eigenvalue of linear transformation x2, nor is its
multiplicity necessarily 1 for the same. Beware when converting
the terbri structure of this word. In fact, the set of all
eigenvalues for a given linear transformation x2 will include
scalar zero (0); therefore, any linear transformation with a
nontrivial set of eigenvalues will have at least two
eigenvalues that may fill in terbri x1 of this word. The
'eigenvalue' of zero for a proper/nice linear transformation
will produce an 'eigenspace' that is equivalent to the entire
vector space at hand. If x3 is specified by a set of vectors,
the span of that set should fully yield the entire eigenspace
of the linear transformation x2 associated with eigenvalue x1,
however the set may be redundant (linearly dependent); the zero
vector is automatically included in any vector space. A
multidimensional eigenspace (that is to say a vector space of
eigenvectors with dimension strictly greater than 1) for fixed
eigenvalue and linear transformation (and generalization
exponent) is degenerate by definition. The algebraic
multiplicity x5 of the eigenvalue does not entail degeneracy
(of eigenspace) if greater than 1; it is the integer number of
occurrences of a given eigenvalue x1 in the multiset of
eigenvalues (spectrum) of the given linear
transformation/square matrix x2. In other words, the
characteristic polynomial can be factored into linear
polynomial primes (with root x1) which are exponentiated to the
power x5 (the multiplicity; notably, not x4). For x4 > x5, the
eigenspace is trivial. x2 may not be diagonalizable. The scalar
zero (0) is a naturally permissible argument of x1 (unlike some
cultural mathematical definitions in English). Eigenspaces
retain the operations and properties endowing the vectorspaces
to which they belong (as subspaces). Thus, an eigenspace is
more than a set of objects: it is a set of vectors such that
that set is endowed with vectorspace operators and properties.
Thus klesi alone is insufficient. But the set underlying
eigenspace x3 is a type of klesi, with the property of being
closed under linear transformation x2 (up to scalar
multiplication). The vector space and basis being used are not
specified by this word. Use this word as a seltau in
constructions such as "eigenket", "eigenstate", etc. (In such
cases, te aigne is recommended for the typical English
usages of such terms. Use zei in lujvo formed by these
constructs. The term "eigenvector" may be rendered as cmima
be le te aigne). See also gei'ai, klesi, daigno