What is an isotropic (spherical) covariance matrix?

Could somebody explain to me in simple terms what an isotropic covariance matrix is? I can’t find anything online.

Answer

A covariance matrix C is called isotropic, or spherical, if it is proportionate to the identity matrix: C=λI, i.e. it is diagonal and all elements on the diagonal are equal.


This definition does not depend on the coordinate system; if we rotate coordinate system with an orthogonal rotation matrix V, then the covariance matrix will transform into VCV=VλIV=VVλI=λI, i.e. will stay the same.

Intuitively, isotropic covariance matrix corresponds to a “spherical” data cloud. A sphere remains a sphere after rotation.

Attribution
Source : Link , Question Author : test , Answer Author : amoeba

Leave a Comment