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Is it possible to compute a distance between two matrices of different rank, and different dimension?

In particular I'm interested in the following case. Suppose $[K]_{ij}=\exp[-(x_i-x_j)^2]$, and let $[\tilde{K}]_{mn} = \vec{k}_m^T\vec{k}_n$, where,

$[k_m^T]_i=\exp[-(x_i-\mu_m)^2]$,

and

$\min_i \{x_i\} \le \mu_1 < \cdots < \mu_{N_s} \le \max_i\{x_i\}$

I'm interested in cases where $N_s=$rank($\tilde{K}$) $<$ rank$(K)=N$.

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    I haven't proven it but I pretty sure both of the matrices I describe are full rank. But thank you I will clarify for the general case2012-05-08

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Answer: You can define a distance between matrices in many ways. For example, convert each matrix to a vector, $(a_{11},\dots, a_{1n}, a_{21},\dots, a_{2n},\dots, a_{nn})$, and add enough zeros to the end of the shorter vector to make them of the same length. Then use the vector norm of the difference. That's a distance all right. Is it of any use? Probably not.

Comment: To come up with a notion of distance that's actually useful, one would need to know that you want to use it for. Do you want to be able to tell that two matrices, even though they are of different size, are somehow "nearly the same". Then you should articulate your idea of "nearness". Given a pair of matrices, when will you say that they are "nearly the same"?