Definition of Elementary Matrices

Question

I'm a bit confused about the definition of elementary matrices which are used to represent elementary row operations on an extended coefficient matrix when doing the Gaussian elimination.

In my lecture at uni, the elementary matrix was defined with the Kronecker delta like so:

$$E_{ij} = (\delta_{ii'} \delta_{jj'})_{1 \leq i', j' \leq m}$$

And a given example was

$$ E_{12} = \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$$

Reading some supplementary literature cleared up how elementary matrices are constructed, as the book defined $E_{ij}$ to be a single-entry matrix and that elementary matrices are constructed by adding a single-entry matrix (or a multiple thereof) to the identity matrix.

For example the elementary matrix representing the row transformation of adding the jth row to he ith row of a matrix $A$ was defined as $$ Q_{ij} = I_m + E_{ij}$$ where it then could be applied to $A$ like so: $$A' = Q_{ij} \cdot A$$ where $A'$ would be the transformed matrix.

This definition makes intuitively way more sense to me than the definition of my university, where we would skip the definition of single-entry matrices and $Q_{ij}$ to just define the same transformation as $A' = (I_m + E_{ij}) \cdot A$.

1. Am I correct to assume that the definition of elementary matrices of my Uni is just wrong?
2. Even if my prof meant to say that $E_{ij}$ is a single-entry matrix; how would one calculate $E_{ij}$ using the two kronecker deltas? I know that the identity matrix can be defined through the kronecker delta like so: $I_m = (\delta_{ij})$ but I have a hard time picturing how a single-entry matrix is calculated through the two kronecker deltas.

Thanks in advance for your help.

Answer 1

Let's answer your question 2 first. The expression with the kronecker deltas is $$ E_{ij} = (\delta_{ii'}\delta_{jj'})_{1\leq i',j' \leq m} $$ This means that the entry at row $i'$, column $j'$ of the matrix $E_{ij}$ is given by $$\delta_{ii'}\delta_{jj'}$$ What value is this expression? Since for any $x,y$ we have $\delta_{xy}=1$ if $x=y$ and $\delta_{xy}=0$ otherwise, we see that $\delta_{ii'}\delta_{jj'}=1$ if and only if both $i=i'$ and $j=j'$ hold, and otherwise the product is zero.

This means that exactly one entry of $E_{ij}$ is nonzero, namely the one at row $i$ and column $j$, as desired. (But I can agree that the notation with the deltas is a bit complicated)

Your question 1 is a bit harder for me to answer, let's see if I understand you correctly:

If the definition given in lecture was that $E_{ij}$ is an elementary matrix, I'd say the definition is wrong. Just as the Wikipedia link you give says, an elementary matrix is a matrix that is one elementary row operation away from the identity matrix. You can express these matrices as $I+kE_{ij}$ for some choice of indices $i,j$ and scalar $k$, or as an identity matrix where you swap two rows (see the examples on the wiki page).

I would guess that since your book explains elementary matrices in this way, maybe you could double-check what was said during the lecture, to be sure you didn't misunderstand? It would be odd if the book and lecturer said different things, so to say.