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I am reading a book which covers the Kronecker delta and shows some examples of how it works. One of the examples confuses me, because it seems to be impossible.

This book uses the notation that a repeated index is a summation over the range of that index. In this case, d and a are both 3x3 matrices.

The statement I have been mulling over is:

[itex]\delta_{jk} a_{ik} = a_{ij}[/itex]

I interpret it this way: since the k is the repeated index, then the summation is over the range of k, like so:

[itex]\sum\limits_{k=1}^3 \delta_{jk} a_{ik} = \delta_{j1} a_{i1} +\delta_{j2} a_{i2} +\delta_{j3} a_{i3} [/itex]

Aren't these column vectors multiplied by column vectors? This makes no sense to me and seems impossible. My professor told me that this statement of mine is wrong, and that all the terms are scalars. I am sure he is right, but I simply do not see how. Could anyone offer some assistance?