I am unable to understand the 2nd definition of row-reduced echelon matrices in

  • Kenneth Hoffman, Ray Kunze, Linear Algebra, 2nd edition 1971.

Attached is the image:

Hoffman Page 12 - 2nd Paragraph

In part a) it says $R_{ij} = 0$ if $j < k_i$ which implies all entries in columns < $k_i$ will be zero. however I would have thought it should be$R_{ij} = 0$ if $j < min(k_i)$ or $R_{ij} = 0$ if $j < k_1$.

Also point b) is not clear because subscript of left side $R_{ik_i}$ and right side $\delta_{ij}$ are different. I mean I was expecting some constraint like $j <= k_i$.

Please help me understand the notation. Even though I am clear what a row reduced Echelon matrix is from the 1st definition but need to understand this notation as this is common across the book.

  • $\begingroup$ Look more closely at condition (b). It’s a bit hard to read in your image, but the subscript is $k_j$, not $k_i$. $\endgroup$ – amd Apr 16 '19 at 18:56

It’s a bit hard to read in the image that you’ve uploaded, but the subscript in condition (b) is $k_j$, not $k_i$, i.e., the condition is really $R_{ik_j}=\delta_{ij}, 1\le i\le r, 1\le j\le r$. This, together with the other conditions, ensures that the only nonzero entry in column $k_i$ occurs in row $i$.

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