If you are given a statement:

"If A has a unique LU-factorization, where L is a lower triangular matrix, then all n leading principal submatrices of A are nonsingular"

How would you show that this is not true?

I wrote a proof like this:

Suppose A is nonsingular with the L U factorization A = LR. Since A is nonsingular it follows that L and R are nonsingular. By definition, we know $A_k=L_k*R_K$. $L_k$ is lower unit triangular and therefore nonsingular. $R_k$ is nonsingular because diagonal entries are amongst the nonzero diagnol entries of R.

This is incorrect. I need to find a counterexample for the beginning statement. But I don't even know where to find a proper countexample. I think the proof I wrote made sense, but perhaps not.

  • $\begingroup$ The $LU$ factorization is never unique, unless you specify some further condition. $\endgroup$ – Martin Argerami Nov 22 '17 at 3:44
  • $\begingroup$ @Martin Argerami well that was my issue, I added additional restrictions. But what would be a counter to show its not unique? $\endgroup$ – Aggrawal Puja Nov 22 '17 at 3:52

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