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==== valsi ====
aigne
==== type ====
fu'ivla
==== creator ====
[[personal/gleki|gleki]]
==== time entered ====
Tue Nov  5 05:52:25 2013
 
== English ==
=== Definition #42482 - Preferred ===
 
==== definition ====
x<sub>1</sub> is an eigenvalue (or zero) of linear transformation/square matrix x<sub>2</sub>, associated with/'owning' all vectors in generalized eigenspace x<sub>3</sub> (implies neither nondegeneracy nor degeneracy; default includes the zero vector) with 'eigenspace-generalization' power/exponent x<sub>4</sub> (typically and probably by cultural default will be 1), with algebraic multiplicity (of eigenvalue) x<sub>5</sub>
==== notes ====

For any eigenvector v in generalized eigenspace <span class="MATH"><i>x</i><sub>3</sub></span> of linear transformation <span class="MATH"><i>x</i><sub>2</sub></span> for eigenvalue <span class="MATH"><i>x</i><sub>1</sub></span>, where I is the identity matrix/transformation that works/makes sense in the context, the following equation is satisfied:  MATH
 $((x_2 - x_1 I)^<a class="undefined" href="../dict/x_4?bg=1;langidarg=2">x_4</a>)v = 0$
 
<span class="MATH">((<i>x</i><sub>2</sub> - <i>x</i><sub>1</sub><i>I</i>)<sup>x<sub>4</sub></sup>)<i>v</i> = 0</span>. When the argument of <span class="MATH"><i>x</i><sub>4</sub></span> is 1, the generalized eigenspace <span class="MATH"><i>x</i><sub>3</sub></span> is simply a strict/simple/basic eigenspace; this is the typical (and probable cultural default) meaning of this word. <span class="MATH"><i>x</i><sub>4</sub></span> will typically be restricted to integer values k > 0. <span class="MATH"><i>x</i><sub>2</sub></span> 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 (<span class="MATH"><i>x</i><sub>1</sub></span>). When filling <span class="MATH"><i>x</i><sub>3</sub></span> and/or <span class="MATH"><i>x</i><sub>4</sub></span>, <span class="MATH"><i>x</i><sub>2</sub></span> and <span class="MATH"><i>x</i><sub>1</sub></span> (in that order of importance) should already be (at least contextually implicitly) specified. <span class="MATH"><i>x</i><sub>3</sub></span> is the set of all eigenvectors of linear transformation <span class="MATH"><i>x</i><sub>2</sub></span>, endowed with all of the typical operations of the vector space at hand. The default includes the zero vector (else the <span class="MATH"><i>x</i><sub>3</sub></span> 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 <span class="MATH"><i>x</i><sub>3</sub></span>. In other words there may or may not be more than one linearly independent vector in the eigenspace <span class="MATH"><i>x</i><sub>3</sub></span> for a given eigenvalue <span class="MATH"><i>x</i><sub>1</sub></span> of linear transformation <span class="MATH"><i>x</i><sub>2</sub></span>. <span class="MATH"><i>x</i><sub>3</sub></span> is the unique generalized eigenspace of <span class="MATH"><i>x</i><sub>2</sub></span> for given values of <span class="MATH"><i>x</i><sub>1</sub></span> and <span class="MATH"><i>x</i><sub>4</sub></span>. <span class="MATH"><i>x</i><sub>1</sub></span> is not necessarily the unique eigenvalue of linear transformation <span class="MATH"><i>x</i><sub>2</sub></span>, 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 <span class="MATH"><i>x</i><sub>2</sub></span> 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 <span class="MATH"><i>x</i><sub>1</sub></span> 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 <span class="MATH"><i>x</i><sub>3</sub></span> is specified by a set of vectors, the span of that set should fully yield the entire eigenspace of the linear transformation <span class="MATH"><i>x</i><sub>2</sub></span> associated with eigenvalue <span class="MATH"><i>x</i><sub>1</sub></span>, 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 <span class="MATH"><i>x</i><sub>5</sub></span> of the eigenvalue does not entail degeneracy (of eigenspace) if greater than 1; it is the integer number of occurrences of a given eigenvalue <span class="MATH"><i>x</i><sub>1</sub></span> in the multiset of eigenvalues (spectrum) of the given linear transformation/square matrix <span class="MATH"><i>x</i><sub>2</sub></span>. In other words, the characteristic polynomial can be factored into linear polynomial primes (with root <span class="MATH"><i>x</i><sub>1</sub></span>) which are exponentiated to the power <span class="MATH"><i>x</i><sub>5</sub></span> (the multiplicity; notably, not <span class="MATH"><i>x</i><sub>4</sub></span>). For <span class="MATH"><i>x</i><sub>4</sub></span> > <span class="MATH"><i>x</i><sub>5</sub></span>, the eigenspace is trivial. <span class="MATH"><i>x</i><sub>2</sub></span> may not be diagonalizable. The scalar zero (0) is a naturally permissible argument of <span class="MATH"><i>x</i><sub>1</sub></span> (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|klesi]] alone is insufficient. But the set underlying eigenspace <span class="MATH"><i>x</i><sub>3</sub></span> is a type of [[klesi|klesi]], with the property of being closed under linear transformation <span class="MATH"><i>x</i><sub>2</sub></span> (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|te]] [[aigne|aigne]] is recommended for the typical English usages of such terms. Use [[zei|zei]] in lujvo formed by these constructs. The term "eigenvector" may be rendered as [[cmima|cmima]] [[be|be]] [[le|le]] [[te|te]] [[aigne|aigne]]). See also [[gei'ai|gei'ai]], [[klesi|klesi]], [[daigno|daigno]]

==== gloss words ====

<ul><li>[[natlang/en/eigen-|eigen- ; prefix; mathematical/physical]]</li><li>[[natlang/en/eigenspace-generalization exponent|eigenspace-generalization exponent ; mathematical]]</li><li>[[natlang/en/eigenspace (generalized)|eigenspace (generalized) ; mathematical; linear transformation, vector space; generalized according to equation aforementioned]]</li><li>[[natlang/en/eigenvalue|eigenvalue ; mathematical; of a square matrix/linear transformation]]</li><li>[[natlang/en/eigenvector|eigenvector ; mathematical; linear transformation/square matrix]]</li><li>[[natlang/en/multiplicity (algebraic) of eigenvalue|multiplicity (algebraic) of eigenvalue ; mathematical; degree of linear terms in characteristic polynomial of the linear transformation/square matrix; useful for Jordan canonical form computations; algebraic mulitplicity]]</li><li>[[natlang/en/self-preserving vector under mapping/transformation|self-preserving vector under mapping/transformation ; mathematical; (perfect preservation not implied: dilation/contraction by scalar, including by scalar zero (0), allowed)]]</li></ul>

==== created by ====
[[personal/gleki|gleki]]
==== vote information ====
3
     
==== time ====
Tue Nov 11 07:50:45 2014

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=== Etymology ===