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A unit lower triangular (ult) matrix $A$ is a matrix whose entries are $a_{ij}=0$ for $j>i$, and $a_{ii}=1$.

I am trying to prove that the inverse of a ult matrix is another ult matrix. (There are different proofs here). I have already proved that the product of ult matrices is also a ult matrix. So I want to use this property by defining $A^{-1}=E_n...E_2E_1I$, where $E_k$ are elementary matrices that correspond to the row operations of the transformation $[A|I] \rightarrow [I|A^{-1}]$. If we can show that $E_k$ are all ult elementary matrices, then the proof is complete. Therefore, the question is:

Considering a ult matrix A, can we transform $[A|I]$ into $[I|A^{-1}]$ only using row operations that correspond to ult elementary matrices?

Community
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1 Answers1

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Hint. Row operations corresponding to unit lower triangular matrices means adding a multiple of a row to a row that is below it. If you think about how you go from $A$ to $I$ in the simplest way, where $A$ is ult, you will see that you are only using operations of this kind.

Let me suggest another proof however. Let $A$ be a ult matrix. Let $V$ be the space of lower triangular matrices. The mapping $X \mapsto AX$ is an endomorphism of the finite-dimensional space $V$. Moreover this mapping is injective, because $A$ is invertible (its columns are linearly independent). Since this mapping is injective, it must also be surjective. Thus $I$ is in its image, meaning that there is a lower triangular matrix $X$ such that $AX = I$, which can only be the inverse of $A$. It is not difficult to see that $X$ must be ult.

Keith
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  • Thank you for the alternative proof but I am following a book (which skips such proofs) and mappings/spaces are in the following chapters. So, I am not currently qualified to understand it. If I can express your hint as a general statement I will be satisfied for now. –  Jul 08 '15 at 14:15
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    @i.ozturk: Follow the hint, and ignore the alternative proof (which is not an answer to the questions). Show that an appropriate composition of row operations (namely, subtractions of multiples of the $1$-st, $2$-nd, etc., $n-1$-st rows from the $n$-th row) turns the $n$-th row into $\left(0,0,\ldots,0,1\right)$. Then, argue by induction. – darij grinberg Jul 08 '15 at 14:22
  • @darijgrinberg Could you please clarify the meaning and purpose of your comment that the alternative method doesn't answer the question? The basic question was "I am trying to prove that the inverse of a ult matrix is another ult matrix." I don't see how it helps the OP to be told to ignore the alternative proof. If it is correct, he himself can decide whether it is useful for him. – Keith Jul 08 '15 at 14:38
  • See the question at the end of the OP; it is not equivalent to the basic question! – darij grinberg Jul 08 '15 at 14:44
  • It was not originally clear whether the OP had latitude to depart from this method, as the first question could be interpreted to be the context of his investigation. In any case, it was perfectly transparent that the alternative proof answered the first question but not the last one; you were stating the obvious in saying this. Your comment appeared to suggest that the alternative proof in fact did not answer even the first question, which was the only thing it ever claimed to do. – Keith Jul 08 '15 at 14:47