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I am currently trying to see if I can decompose a matrix M of rank r, into the sum of two matrices $N_1$ and $N_2$, of rank respectively $r_1$ and $r_2$ such that $r_1,r_2 < r$.

According to this post, $$ rank(N_1 + N_2) \leq rank(N_1)+rank(N_2) $$

Let say that in my case, I would be happy only if the equality hold, so that the new decomposition is made of somehow more simple matrices.

I know that this decomposition is at the root of methods like jacobi or gauss seidel, but in this case, the decomposition is trivial. How one would choose $N_1$ and $N_2$ while imposing an equidistribution of the "complexity" (rank) among them ?

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Tobbey
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  • For the equality $r_1 + r_2 = r$, it is enough to compute the singular value decomposition. – Alex Silva Sep 14 '16 at 12:05
  • Eigen decomposition indeed provide a sum of r rank 1 matrices, but how can I be sure that if I use, let say, the sum of r/2 rank 1 matrices for $N_1$ and the sum of the other $r/2$ matrices for $N_2$, $r_1$ and $r_2$ will be balanced ? – Tobbey Sep 14 '16 at 14:04
  • What do you mean by "balanced"? – Alex Silva Sep 14 '16 at 14:06
  • The rank of $N_1$ and the rank of $N_2$ should be close to each other. – Tobbey Sep 14 '16 at 14:30
  • The SVD ensures the orthogonality of the decomposable matrices. Thus, all rank-1 matrices are orthogonal to one another. There is no problem to "balance" the matrices $N_1$ and $N_2$. – Alex Silva Sep 14 '16 at 17:44
  • Ok that is a property of svd I forgot, be cause I use to work with symmetric matrices and eigenvalues where this always hold. Then thank you I would accept your answer if you write one – Tobbey Sep 14 '16 at 19:33

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