Euclidean distance of a tensor?

Question

The definition of the euclidean norm of matrix is clearly defined:

$$|A|= \sqrt{\sum_{ij} a_{ij}^2}$$

But does it work, if I have a tensor (x,y,$depth>1$) which has a depth $>1$ ($depth == 1$ would be a matrix).

Answer 1

The double-dot product is the square of the 2-norm for any second-order tensor $\boldsymbol{S}$. Using Einstein's summation convention, its value may be written \begin{aligned} {\|\boldsymbol{S}\|_2}^2 &= \boldsymbol{S} : \boldsymbol{S} \\ &= S_{ij} S_{ij} \\ &= \sum_{ij} {S_{ij}}^2 \, , \end{aligned} where $S_{ij}$ are components of $\boldsymbol{S}$. One may proceed similarly for third-order tensors $\boldsymbol{T}$ by setting \begin{aligned} {\|\boldsymbol{T}\|_2}^2 &= T_{ijk} T_{ijk} \\ &= \sum_{ijk} {T_{ijk}}^2\, . \end{aligned}

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Note: zeroth-order tensor $\sim$ scalar, first-order tensor $\sim$ vector, second-order tensor $\sim$ matrix.