# 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.

A covariance matrix $\mathbf C$ is called isotropic, or spherical, if it is proportionate to the identity matrix: $$\mathbf C = \lambda \mathbf I,$$ 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 $\mathbf V$, then the covariance matrix will transform into $$\mathbf V^\top \mathbf C \mathbf V = \mathbf V^\top \cdot \lambda \mathbf I \cdot\mathbf V = \mathbf V^\top \mathbf V \cdot \lambda \mathbf I = \lambda \mathbf I,$$ i.e. will stay the same.

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

• What if the variables can be rotated to get to $\lambda \mathbf I$ covariance matrix? Commented Mar 30, 2016 at 18:27
• @Aksakal See update. Commented Mar 30, 2016 at 18:49
• +1. But curiously, a completely different definition of "isotropic" also applies to $C$ because--as is usual with covariance matrices--it represents a quadratic form on a real vector space. But in this other sense, the only isotropic covariance matrix is the zero matrix!
– whuber
Commented Mar 30, 2016 at 20:11
• @whuber Interesting! I did not remember that there exists a notion of "isotropic" quadratic forms. But reading the definition now, wouldn't any covariance matrix with at least one zero eigenvalue be "isotropic" in that sense? Commented Mar 30, 2016 at 20:15
• You're right--I mis-specified the quantifier. By definition, an isotropic quadratic form has at least one nonzero isotropic vector (rather than all vectors being isotropic).
– whuber
Commented Mar 30, 2016 at 20:23

The covariance is only a function of $|x - x'|$. You can find a definition there.

Edit: sorry I misread, for matrix, the right answer is amoeba's one.

• The questions asks about covariance matrix. Of course a matrix can be seen as a function, but I guess this requires some elaboration for the OP. Commented Mar 30, 2016 at 18:24