Matrix Identify involving Diagonal Matrix

[hotvette]

If V is diagonal, it is easy to show:

$(V + V^{-1})^{-1} = V(V^2 + I)^{-1}$

by multiplying both sides by:

$(V + V^{-1})$

But, I'm wondering if there is a way to derive the RHS from the LHS. Since diagonal matrices behave like scalars, I used a scalar analogy:

$(x + 1/x)^{-1} = ((x^2 + 1)/x)^{-1} = x(x^2 + 1)^{-1}$

But I'd like to show it using matrix terminology. If there is something simple, I'm missing it. Any ideas?

 

[morphism]

V doesn't even have to be diagonal, only invertible (and we also need V^2+I to be invertible - if we're working with real matrices, this is automatic).

$$(V + V^{-1})^{-1} = ((V^2 + I)V^{-1})^{-1} = V(V^2 + I)^{-1},$$

where in the middle step I'm pulling out the "common factor" $V^{-1}$, and in the last step I'm using the fact that $(AB)^{-1} = B^{-1}A^{-1}$.

 

[hotvette]

I knew it had to be simple. Just didn't see it...

Thanks.