The summation convention for Tensor Notation says, that we can omit the summation signs and simply understand a summation over any index that appears twice. So consider a 3X3 matrix A whose elements are denoted by aij, where i and j are indices running from 1 to 3. Now consider the multiplication aiαaiβ. 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, aiαaiβ=δαβ Where δ is the dirac delta function. Now what if α=β? According to the above equation, aiαaiβ should equal 1 since δαβ=1 for α=β. But if we write it as aiαaiα, 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??