# The logarithm of a symmetric matrix and non-symmetric matrix.

As I was reading a paper, I came across a formulation for the mixing rule of two matrices, where each matrix represents the deformation of a body in a particular phase. It was written that for each pair of matrices $$\mathbf{A}$$ and $$\mathbf{B}$$ when the matrices are symmetric we use the following logarithmic mixing rule,

$$\mathbf{C}=\eta \log \mathbf{A}+(1-\eta)\log \mathbf{B}$$

where $$\mathbf{C}$$ is the mixing matrix and $$\eta$$ is the mixing factor.

On the other hand, when the matrices are not symmetric we use the following linear rule for mixing them,

$$\mathbf{C}=\eta \mathbf{A}+(1-\eta)\mathbf{B}$$.

Now, regardless of the theory behind the above formulations, the question is that is there something special about the symmetric matrices that allow us to use the logarithmic mixing rule for them that the non-symmetric matrices do not possess.