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1.) Let $A$ be a square matrix of order $n.$ Define $V_k(A)$ as $k\times k$ upper left submatrix $A.$ Suppose $A$ has $LU$ decomposition, is it true that $V_k(A)$ is invertible for all $k \ ?$

2.) I read that if $A$ is invertible, then it has $LU$ decomposition if and only if all its leading principal minors are non-zero. May I know what is a principal leading minor? Examples will be greatly appreciated!

Thank you.

Alexy Vincenzo
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  • 2.) These are precisely the $V_k\left(A\right)$. See https://math.stackexchange.com/questions/2951461/lu-factorization-of-a-nonsingular-matrix-exists-if-and-only-if-all-leading-princ/2955663#2955663 for proofs of the result you are referring to. – darij grinberg Sep 19 '19 at 16:15
  • 1.) No, because $A$ may be the zero matrix. In general (i.e., if you don't require $A$ to be invertible), I don't know of a good criterion for when $A$ has an LU decomposition. – darij grinberg Sep 19 '19 at 16:17
  • @darijgrinberg: Thank you. If we require $A$ be to non-zero as well, hence, if $A$ had LU decomposition, does it mean $V_k(A)$ is invertible for all $k?$ – Alexy Vincenzo Sep 19 '19 at 16:24
  • Nope, you need $A$ to be invertible for that conclusion. There is a lot of room between fully zero and invertible. – darij grinberg Sep 19 '19 at 16:25
  • @darijgrinberg: Thank you for the advice. Is there an example of a square matrix $A$ that has $LU$ decomposition but $V_k(A)$ is not invertible for some $k ?$ – Alexy Vincenzo Sep 19 '19 at 16:33
  • You can easily come up with one: Just multiply a lower-triangular and an upper-triangular matrix whose diagonal entries are not all nonzero. – darij grinberg Sep 19 '19 at 16:34

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