I am trying to compute the Moore- Penrose inverse of a given $m\times n$ matrix $A$. I did convergence analysis then I came across to the following condition.

For convergence of my method: $\max\mid 1 - \lambda_\max(X_0 A))\mid <1$ where $X_{0}$ is an $n\times m$ matrix and is the initial approximation for my sequence of iterations to compute Moore- Penrose inverse and $\lambda$ stands for eigenvalue of a matrix.

My question is how could I choose $X_{0}$ so that this condition $\max\mid 1 - eigenvalue(X_0 A))\mid <1$ holds true.

I know one possibility that is to choose $X_{0} = \alpha A^t$ and $0<\alpha<2/\lambda_\max(AA^t)$, and for that

But I need any other alternate possibility. I need help with this. please help me. I would be very much thankful.


1 Answer 1


More general solution includes $X_0=\alpha_{ij}$. You will then find the roots of the characteristic function of $\alpha_{ij}A$. Here you need to choose $\alpha_{ij}$ matrix such that your condition is satisfied. There are quite many possible solutions in this case. However you might be interested in the minimum of the maximum of the eigen values of this matrix. To find such if your matrix $A$ is deterministic $\alpha_{ij}$ should be of random nature. Ideally among all possible $\alpha_{ij}A$ which has the same enegy, the one which has a random nature will minimize the maximum eigen value. Because eigen values are related to the spectrum of a matrix and random noise has a flat spectrum implying the equality of eigen values. We know that eigen values can be ordered in non increasing order. Therefore equal eigen values indicate a minimax solution for this problem.

I hope this helps

  • $\begingroup$ I am sorry I didn't understand what you mean by $\alpha_{i,j}$ should be of random nature. Could you please explain? $\endgroup$
    – Srijan
    Aug 10, 2012 at 9:07
  • $\begingroup$ It should be a type of randomly generated matrix.If you find the roots of the characteristic function in a parametric form,namely in terms of $\alpha_{ij}$,and given the constraint $\sum_{ij}\alpha{ij}^2=T$,you can solve the system such that $\lambda_0,\lambda_1,...,\lambda_N$ are as close as possible to each other.This can be done for example via minimizing such a metric $(\lambda_0-\lambda_1)^2+(\lambda_1-\lambda_2)^2...,(\lambda_{N-1}-\lambda_N)^2$.After you solve this problem you will get $\alpha_{ij}$.There might be multiple solutions but any solution will give you a random like matrix $\endgroup$ Aug 10, 2012 at 10:11

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