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So consider a 3X3 matrix A whose elements are denoted by a

_{ij}, where i and j are indices running from 1 to 3.

Now consider the multiplication a

_{iα}a

_{iβ}.

Using the summation convention described above, the summation here would be over the index i since it occurs twice.

Now if the matrix A is an orthogonal matrix, then it has the property that elements of any row or column can be thought of as components of a vector whose magnitude is 1, and that they are all mutually orthogonal.

So, a

_{iα}a

_{iβ}=δ

_{αβ}

Where δ is the dirac delta function.

Now what if α=β?

According to the above equation, a

_{iα}a

_{iβ}should equal 1 since δ

_{αβ}=1 for α=β.

But if we write it as a

_{iα}a

_{iα}, by summation convention, this means a summation over both i and α(or β).

First summing over α, this means multiplication of each element of the i th row with itself.

This will equal 1, as a result of A being orthogonal.

Now summing over i, we'll get i*1=i.

Also, if we had summed over i first and then α, we would have got α*1=α.

Where am I going wrong??