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 () which on multiplying vector *x* results in the same vector as that obtained by above matrix multiplication. Mathematically,

Here, refers to Eigen Vector and refers to Eigen value.

References:

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

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