condition number of a product of matrices Reading about the condition number of a matrix I encountered the equality $\kappa (M^{-1}A) = \kappa(M^{-1/2}AM^{-1/2})$ for some invertible matrices $A,M\in\mathbb R^{n\times n}$ and $A$ symmetric positive definite.
Why does this hold? And can one say in general that $M^{-1}A = M^{-1/2}AM^{1/2}$?
 A: You wrote in your comment that

The full line says: $\kappa(M^{-1}A)=\kappa(M^{-1/2}AM^{-1/2})=\frac{\lambda_\max(M^{-1/2}AM‌​^{-1/2})}{\lambda_\min(M^{-1/2}AM^{-1/2})}$ whereas $\lambda_\max$ and $\lambda_\min$ are the biggest and smallest eigenvalues. $M$ is only given as invertible.

From what it says, it seems that actually,


*

*$M$ is positive definite,

*the condition number is defined using the operator 2-norm, and

*$M$ commutes with $A$.


If the above conditions are satisfied, the equality does hold because $M^{-1}A=M^{-1/2}AM^{-1/2}$ and for positive definite matrices, the ordered singular values coincide with the ordered eigenvalues.
You should read the paper again to see if there is any implication or passing mention that the above conditions are satisfied. If not, then either there are some other unspecified conditions, or the authors are simply wrong, because the equality does not hold in general. For a counterexample, suppose $\kappa(X):=\|X^{-1}\|_2\|X\|_2$ and consider
$$
M^{-1/2}=\pmatrix{2&1\\ 1&1},\ M^{-1}=\pmatrix{5&3\\ 3&2},\ A=\pmatrix{1&0\\ 0&2}.
$$
In this counterexample, only condition 3 is violated. One can verify that
$$\kappa(M^{-1}A)\approx 42.98 \ne 38.47\approx\kappa(M^{-1/2}AM^{-1/2})=\frac{\lambda_\max(M^{-1/2}AM‌​^{-1/2})}{\lambda_\min(M^{-1/2}AM^{-1/2})}.$$
