Distance between a matrix and it inverse? It is know that the product of a matrix $A$ and its inverse is identity matrix. Is there a distance measure between inverse of a matrix and another matrix. For example if $A$ is a matrix I want to find a distance for a matrix $B$ to inverse of $A$. I know the possible norms of a matrix and hence possible distance between two matrices. But I am wondering is there a very specific measure for Inverse? What I am looking for is a to analyse a measure which deviate fast from the true inverse. What I mean by deviate fast is that a measure which will clearly distinguish the true Inverse from the other matrices.
 A: No single norm is "preferred" for computing $\Vert A^{-1} - B \Vert$. You might choose a norm based on your particular application, or to achieve a certain result.

However, note that $\mathbb{C}^{n \times n}$ forms a finite dimensional vector space, and hence all  norms on $\mathbb{C}^{n \times n}$ are equivalent. That is, if $|\cdot|$ and $\Vert \cdot \Vert$ are norms, there exist positive constants $c$ and $c^\prime$ such that
$$c | X | \leq \Vert X \Vert \leq c^\prime |X| \text{ for all } X \in \mathbb{C}^{n \times n}.$$
A consequence of this is that if you have a sequence $\{B_n\}$ which converges to $A^{-1}$ in one norm, it converges in any norm. That is to say (very roughly)...

if $B_n$ gets close to $A^{-1}$ in one norm, it gets close to $A^{-1}$ in any other norm.

This might inadvertently answer your question.

Note: I used $\mathbb{C}$ for simplicity, but you do not have to.
A: I think, there is not a particular distance for inverse pairs, which is not one of the norms you mentioned. You could consider the Frobenius distance between two matrices $A$ and $B$. Of course, there are many other possibilities: if the matrices are $\mathbf{A} = (a_{ij})$ and $\mathbf{B} = (b_{ij})$, then some examples are:
$$
d_1(\mathbf{A}, \mathbf{B}) = \sum_{i=1}^n \sum_{j=1}^n |a_{ij} - b_{ij}|
$$
$$
d_2(\mathbf{A}, \mathbf{B}) = \sqrt{\sum_{i=1}^n \sum_{j=1}^n (a_{ij} - b_{ij})^2}
$$
$$
d_\infty(\mathbf{A}, \mathbf{B}) = \max_{1 \le i \le n}\max_{1 \le j \le n} |a_{ij} - b_{ij}|
$$
$$
d_m(\mathbf{A}, \mathbf{B}) = \max\{ \|(\mathbf{A} - \mathbf{B})\mathbf{x}\| : \mathbf{x} \in \mathbb{R}^n, \|\mathbf{x}\| = 1 \}
$$
