I have two matrices ${\bf H_1}$ and ${\bf H_2}$, Id like to create a metric that describes how close there singular values are assume ${\bf H_1}$ has eigen values $\lambda_{11}$ and $\lambda_{12}$ in decreasing order and ${\bf H_2}$ has eigen values $\lambda_{21}$ and $\lambda_{22}$ in decreasing order,, can i say a good metric is
$$\frac{|\lambda_{11}- \lambda_{12}| + |\lambda_{21}- \lambda_{22}| }{\lambda_{11}+\lambda_{12}}$$
And if it makes sense does it mean the closer this metric is to zero the more the matrices' eigenvalues match? Also is this metric between 0 and 1?
THANKS