The term eigenvector localization applies to an eigenvector where the majority of its length is contributed by a small number of entries. This means that majority of its entries are zero or close to...

The term eigenvector localization applies to an eigenvector where the majority of its length is contributed by a small number of entries. This means that majority of its entries are zero or close to zero. This phenomenon is well known in several scientific applications such as quantum mechanics,
 data, and astronomy. If the eigenvector is normalized, one measure of this property is the inverse participation ratio
 that is defined as

The larger the value of IPR, the more localized the eigenvector.

 If the eigenvector values are equally distributed throughout its indices, show that the IPR is
 n

 Show that if the eigenvector has only one nonzero entry, its IPR is 1.

 If
 is an integer, the following statements generate a random symmetric tridiagonal matrix. Create a
random symmetric tridiagonal matrix and plot the eigenvector number against IPR

_i
 What do you observe