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| valsi |
aigne |
| type |
fu'ivla |
| creator |
gleki |
| time entered |
Tue Nov 5 05:52:25 2013 |
Examples
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English
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Definition #42482
- Preferred
[edit]
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| 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:
((x2 - x1I)x4)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
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| gloss words |
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| created by |
gleki |
| vote information |
3
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| time |
Tue Nov 11 07:50:45 2014 |
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[View
Comments For This Definition] |
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Examples
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| krtisfranks | Based entirely on the prefix "eigen-" from German meaning "own, inherent, proper, possessive, particular, special"; converted to Lojban orthography and then having the final two phonemes exchanged so as to morphologically and phonotactically become a valid fu'ivla in a simple manner. Usage as in English mathematical and physics contexts. See also: {jinzi}. | [edit] |
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