## Eigenvector vs Eigenvalue – What’s the difference?

Eigenvector vs Eigenvalue - What's the difference?
As nouns the difference between eigenvector and eigenvalue is that eigenvector is (linear algebra) a vector that is not rotated under a given linear transformation; a left or right eigenvector depending on context while eigenvalue is (linear algebra) the change in magnitude of a vector that does not change in direction under a given linear transformation; a scalar factor by which an eigenvector is multiplied under such a transformation.

## English

### Noun

• (linear algebra) A vector that is not rotated under a given linear transformation; a left or right eigenvector depending on context.
• (physics, engineering) A right eigenvector; a nonzero vector
x

such that, for a particular matrix

A

,

A x = lambda x

for some scalar

lambda

which is its eigenvalue and an eigenvalue of the matrix.

• #### Synonyms

* latent vector, proper vector

* Mathworld article on eigenvectors
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## English

### Noun

(en noun )

• (linear algebra) A scalar,
lambda!

, such that there exists a vector

x

(the corresponding eigenvector) for which the image of

x

under a given linear operator

rm A!

is equal to the image of

x

under multiplication by

lambda

; i.e.

{rm A} x = lambda x!
• ”The eigenvalues

lambda!

of a square transformation matrix

rm M!

may be found by solving

det({rm M} – lambda {rm I}) = 0!

.

#### Usage notes

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by

{rm M} x = lambda x!

for some right eigenvector

x!

. Left eigenvalues, charactarised by

y {rm M} = y lambda!

also exist with associated left eigenvectors

y!

. For commutative operators, the left eigenvalues and right eigenvalues will be the same, and are referred to as eigenvalues with no qualifier.

#### Synonyms

* characteristic root
* characteristic value
* eigenroot
* latent value
* proper value