What is an Eigenvalue and an Eigenvector?
In: Mathematics
In the language of vectors and matrices; A geometric vector is basically something with a length and direction. You can transform vectors by multiplying matrices by them, for example a rotation of 180 degrees in 2D can be represented by the matrix
|-1|0|
|:-|:-|
|0|-1|
If you multiply any vector by this matrix then it will rotate it around 180 degrees.
Now most matrices aren’t that simple and they will do a combination of operations, changing the length and direction of vectors in complex ways.For a given matrix an eigenvector is one which doesn’t have its direction changed, i.e only its length is scaled when you multiply it by that matrix. The eigenvalue is how much that length is scaled.
For example if we have some matrix M and a vector (3,5). Then say we multiply M*V and get a new vector (6,10) which = 2*(3,5). This means V is an eigenvector of M with an eigenvalue of 2, applying M to V has just scaled it by a factor 2 without changing its direction.
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