# Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar.

I think finding the distance between two given matrices is a fair approach since the smallest Euclidean distance is used to identify the closeness of vectors.

I found that the distance between two matrices ($A,B$) could be calculated using the Frobenius distance $F$:

$$F_{A,B} = \sqrt{trace((A-B)*(A-B)')}$$

where $B'$ represents the conjugate transpose of B.

I have the following points I need to clarify

• Is the distance between matrices a fair measure of similarity?
• If distance is used, is Frobenius distance a fair measure for this problem? any other suggestions?
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There are many ways to measure the "distance" between two matrices (just as there are many ways to measure the distance between two vectors). Without some more information, it's impossible to say which one is best for you. –  bubba Sep 28 '13 at 12:40
@bubba I just want to find the closest matrix to a give matrix numerically. I'm creating a closest match retriever for a given matrix. If you can let me know the other possible methods you know for distance measures that would be a great help. –  Synex Sep 28 '13 at 15:51
Provided that you "just want to compute something", a good norm is the norm for which the problem can be easily solved. This is often the case for the Frobenius norm. –  Algebraic Pavel Sep 29 '13 at 11:46
As I said, there are many ways to measure the "distance" between two matrices. If the matrices are $\mathbf{A} = (a_{ij})$ and $\mathbf{B} = (b_{ij})$, then some examples are: $$d_1(\mathbf{A}, \mathbf{B}) = \sum_{i=1}^n \sum_{j=1}^n |a_{ij} - b_{ij}|$$ $$d_2(\mathbf{A}, \mathbf{B}) = \sqrt{\sum_{i=1}^n \sum_{j=1}^n (a_{ij} - b_{ij})^2}$$ $$d_\infty(\mathbf{A}, \mathbf{B}) = \max_{1 \le i \le n}\max_{1 \le j \le n} |a_{ij} - b_{ij}|$$ $$d_m(\mathbf{A}, \mathbf{B}) = \max\{ \|(\mathbf{A} - \mathbf{B})\mathbf{x}\| : \mathbf{x} \in \mathbb{R}^n, \|\mathbf{x}\| = 1 \}$$ I'm sure there are many others. If you look up "matrix norms", you'll find lots of material. And if $\|\;\|$ is any matrix norm, then $\| \mathbf{A} - \mathbf{B}\|$ gives you a measure of the "distance" between two matrices $\mathbf{A}$ and $\mathbf{B}$.
Or, you could simply count the number of positions where $|a_{ij} - b_{ij}|$ is larger than some threshold number. This doesn't have all the nice properties of a distance derived from a norm, but it still might be suitable for your needs.