# Thread: tensor and einstein summation notation

1. ## tensor and einstein summation notation

Hello,

I have some basic problems with tensors and Einstein summation notation. As I understand, if a suffix appears twice in a tensor then summation over repeated indices is assumed, i.e.
$\displaystyle S_{ii} = S_{11} + S_{22} + S_{33}$,
where $\displaystyle S$ is a second order tensor and $\displaystyle i$ is the repeated suffix. Is $\displaystyle S_{ii}$ indirectly a zeroth order tensor, i.e. a scalar?

Einstein summation convention also states that a suffix cannot appear more than twice in an expression. I came across an expression for fourth order tensor $\displaystyle T_{iijj}$, where it is mentioned that $\displaystyle T_{iijj}$ is a scalar. Why is $\displaystyle T_{iijj}$ a zeroth order tensor, i.e. a scalar? I would not expect that since two different suffices are used.

2. Originally Posted by amrasa81
Hello,

I have some basic problems with tensors and Einstein summation notation. As I understand, if a suffix appears twice in a tensor then summation over repeated indices is assumed, i.e.
$\displaystyle S_{ii} = S_{11} + S_{22} + S_{33}$,
where $\displaystyle S$ is a second order tensor and $\displaystyle i$ is the repeated suffix. Is $\displaystyle S_{ii}$ indirectly a zeroth order tensor, i.e. a scalar?

Einstein summation convention also states that a suffix cannot appear more than twice in an expression. I came across an expression for fourth order tensor $\displaystyle T_{iijj}$, where it is mentioned that $\displaystyle T_{iijj}$ is a scalar. Why is $\displaystyle T_{iijj}$ a zeroth order tensor, i.e. a scalar? I would not expect that since two different suffices are used.
When you sum over a repeated suffix, it reduces the order of the tensor by 2. For the fourth order tensor $\displaystyle T_{iijj}$, when you sum over i the order goes down to 2, then when you sum over j it goes down to 0, so you are left with a scalar.

For a simple example, a square matrix with entries $\displaystyle A_{ij}$ is a second-order tensor. The sum $\displaystyle A_{ii}$ is the trace of the matrix, which is a scalar.