How Do You Spell EIGENSPACE?

Pronunciation: [ˈa͡ɪd͡ʒənspˌe͡ɪs] (IPA)

The word "eigenspace" is commonly used in mathematics and refers to the set of all eigenvectors of a linear transformation that corresponds to a given eigenvalue. The IPA phonetic transcription of this word is /ˈaɪʒənspeɪs/, which breaks down as follows: "aɪ" representing the diphthong, "ʒ" representing the voiced postalveolar fricative sound, "n" representing the alveolar nasal, "s" representing the voiceless alveolar fricative sound, and "peɪs" representing the stressed pronunciation of "space". The spelling of "eigenspace" is consistent with the common conventions of English phonetics.

EIGENSPACE Meaning and Definition

1. An eigenspace is a fundamental concept in linear algebra that refers to a vector space corresponding to a specific eigenvalue of a given linear operator or matrix. In simpler terms, it is a collection of vectors that are multiplied by a constant factor when operated on by a particular matrix.

   To explain further, let's consider a linear transformation T on a vector space V. If there exists a non-zero vector v in V such that T(v) is proportional to v, then v is called an eigenvector of T, and the scalar proportionality factor is named eigenvalue. Consequently, the set of all eigenvectors corresponding to a unique eigenvalue λ form the eigenspace associated with λ. Mathematically, this can be represented as the null space (also called kernel) of the linear transformation (T - λI), where I represents the identity matrix.

   The eigenspace is characterized by important properties. Firstly, it is always a subspace of the vector space V, meaning it is closed under vector addition and scalar multiplication. Secondly, the dimension of the eigenspace is the multiplicity of the associated eigenvalue.

   Eigenspaces find valuable applications in various fields, including physics, engineering, and computer science. They aid in understanding the behavior and properties of linear transformations and matrices, enabling the analysis and manipulation of complex systems with relative ease.

Common Misspellings for EIGENSPACE

• eigenstae

Etymology of EIGENSPACE

The word "eigenspace" is derived from the combination of two terms: "eigen" and "space".

1. "Eigen" is a German word meaning "characteristic" or "own". It comes from the German word "eigene", which means "own" or "proper".

2. In mathematics, an "eigenvalue" represents a characteristic value associated with a linear operator or matrix. The term "eigenvector" is used to describe the vector that remains unchanged in direction (but may be scaled) when multiplied by a given matrix.

3. The term "space" refers to an abstract mathematical concept of a set of elements or objects. In linear algebra, a "vector space" refers to a collection of vectors that satisfy certain properties.