3

Given any (singular or non-singular) matrix, is there a standard measure for how close the matrix is close to diagonal dominance? For example this matrix: \begin{pmatrix} 5 &2 &2 \\ 5 &1 &1 \\ 5 &0 &0 \end{pmatrix}

I can't use permutation on the columns to make it diagonal dominant. Is there a measure for sorting the columns and determine it to be as close to diagonal dominant as possible?

Xander Henderson
  • 32,453
Yair Yakoby
  • 43
  • Good question. What if you used $\sum_i H_0\left( s(i) \right)s(i)$, where $s(i)=-|a_{ii}| + \sum_{j\ne i} |a_{ij}| $ and $H_0$ is the Heaviside step? So then it is zero if it is DD, and increasingly positive in the non DD case. – user3658307 Jun 11 '17 at 00:42
  • it's a nice idea, not sure if i can think of a way of proving optimality, i think my main problem is actually defining what is optimal.. – Yair Yakoby Jun 12 '17 at 07:28
  • Hmm well you could also look at $E=\min_{A \in P} \sum_i H_0(s(i)) s(i)$ ie the minimum over all column permutations $P$ of the matrix. This might be harder to prove things about though. Then it's DD iff $E=0$. – user3658307 Jun 12 '17 at 12:09
  • yeah that is what originally intended when i asked the question, find a measure and then use permutation to find the minimum for a given matrix... i'm just not sure that the minimum would give better results then something else and of course i can think of lots of functions to minimize... – Yair Yakoby Jun 12 '17 at 13:49
  • 2
    Possible duplicate of Measure of "how much diagonal" a matrix is – kjetil b halvorsen Aug 14 '18 at 08:13

0 Answers0