Rotation matrix: "sum of squared elements along a row/column is unity: dot product of two different rows is zero"?

Question

My book talking about a rotation matrix says:

" If we take the sum of the squares of elements along a row or along a column , we get unity. But if we take the “dot product” of a row with a different row (or a column with a different column),we get zero: $$R_{ij}R_{ik} = R_{ji}R_{ki} = \delta_{jk}$$"

How is the sentence correct? By definition of the Kronecker delta we know that it is $1$ when $j=k$ and $0$ when $j \neq k$. Hence we have $$R_{ij}R_{ij} = 1$$ but by Einstein summation convention we know that $$R_{ij}R_{ij}=\sum_{i=1}^3 \sum_{j=1}^3R_{ij}R_{ij} = R_{11}^2+R_{12}^2+R_{13}^2+R_{21}^2+R_{22}^2+R_{23}^2+R_{31}^2+R_{32}^2+R_{33}^2$$ which is different from the sum of the square element along a column (or a row). This is actually the sum of the squares of all elements.

Is it wrong or am I wrong?

Answer 1

I guess in $R_{ij}R_{ik} = \delta_{jk}$ the summation is assumed to be on the repeated index, $i$. For the second $R_{ji}R_{ki} = \delta_{jk}$ the summation index is only i. Now $R_{ij}R_{ij}$ is summed only by $i$ or $j$.

Answer 2

$R_{ij}R_{ij}$ does not mean the same thing as $R_{ij}R_{ik}$ with $j$ and $k$ set to the same value. You’ve introduced a second summation that doesn’t exist in the original expression by replacing the index $k$ in the second expression by the index $j$. To put it another way, the summation convention says that the second expression means $\sum_iR_{ij}R_{ik}$ no matter what values $j$ and $k$ have. The first expression instead means $\sum_i\sum_jR_{ij}^2$, which is quite different.