Eigen vectors and Eigen values

A point x in a two-dimensional space represents a vector because it has a magnitude and a direction with respect to the center (0, 0). A scalar multiplication of x represents another vector which lies on the same line (elongated or scaled down) as that of vector x.  When we multiply vector x with a matrix A, it again results in a vector but now the resultant vector will be either in the same previous direction as that of x or in a new direction. Also, the resultant vector will get either scaled up or down. If the resultant vector lies in the same direction then we say vector x is Eigen vector of matrix A, otherwise, it is not an Eigen vector. A 96 seconds youtube video explains the same concept visually.

Corresponding to Eigen vector, we too get a scalar value (\lambda) which on multiplying vector x results in the same vector as that obtained by above matrix multiplication. Mathematically,

A*x =   \lambda*x

Here, x refers to Eigen Vector and \lambda refers  to Eigen value.

 

References:

  1.  https://www.youtube.com/watch?v=wXCRcnbCsJA
  2. http://blog.stata.com/2011/03/09/understanding-matrices-intuitively-part-2/
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