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Let $S$ be a system of equations with $n$ variables and $n$ equations, and let $H$ be its associated matrix. Let $H'$ be the upper triangular matrix that is equivalent to $H$. Then there is a theorem that states that $S$ has a unique solution iff $H'_{jj}\neq0$ for $i \in [1, n]$. In other words, if each diagonal element of $H'$ is not null.

This seems to implicity assume that upper triangularization is unique. Otherwise, if some $H'_{jj}=0$, how could we know there isn't some other upper triangular matrix equivalent to $H$ without this property?

How can one prove there is a unique upper triangular matrix $H'$ derivable from $H$ via the triangularization method?

lafinur
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  • Does this answer your question? Is matrix diagonalization unique? – Chris Eagle Dec 09 '22 at 00:08
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    "upper diagonal"? Did you mean "upper triangular"? It looks like you are confusing triangular with diagonal. Please, clarify your question, which one do you have in mind? – jjagmath Dec 09 '22 at 00:08
  • Sorry, I meant upper triangular. I'll fix it. – lafinur Dec 09 '22 at 00:12
  • Does "equivalent" mean "row-equivalent"? (That is, is the idea that $H'$ can be obtained from $H$ by invertible row operations?) – Misha Lavrov Dec 09 '22 at 00:16
  • I have edited the question fixing the fact that I was confounding the term "upper triangularization" and "diagonalization" (this comes from translation, I'm not a native English speaker, I apologize). – lafinur Dec 09 '22 at 00:32
  • My question is if, given a matrix $H$, there is only one upper triangularization $H'$ of $H$? – lafinur Dec 09 '22 at 00:32
  • @ChrisEagle No, since it is about diagonalization, not upper triangularization. As I said, that is my bad. Thank you still. – lafinur Dec 09 '22 at 00:33
  • Note that physicists‘ “equivalence” means unitary similarity ($A\sim B$ iff $A=UBU^\dagger$ for some unitary $U$), which has a narrower meaning than linear algebraists‘ “equivalence” ($A\sim B$ iff $A=PBQ$ for some invertible $P$ and $Q$), which means equality of ranks. Your question clearly concerns about some sort of similarity, but do you mean mere similarity ($A\sim B$ iff $A=SBS^{-1}$ for some invertible $S$ that is not necessarily unitary) or unitary similarity? – user1551 Dec 09 '22 at 07:02

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