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For example:

$$ X = \begin{bmatrix} 1 & -1 & 0 & 0 \\ -1 & 0 & 5 & 3 \\ -2 & 1 & 0 & 0 \\ 0 & 1 & 0 & 2 \end{bmatrix} $$

How to know if it is a (0,2) tensor, a (2,0) tensor or a (1,1) tensor? What is the difference between these 3 forms?

Qmechanic
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Firestar-Reimu
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  • Related: https://physics.stackexchange.com/q/105347/2451 , https://physics.stackexchange.com/q/119126/2451 , https://physics.stackexchange.com/q/20437/2451 and links therein. – Qmechanic Aug 17 '23 at 08:57
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    you can’t. A matrix by itself is pretty meaningless; it’s just an array of numbers. Also, please don’t confuse tensors for their components which then get arranged into a matrix. – peek-a-boo Aug 17 '23 at 10:13
  • $X_{ij}, X^{ij}, X^i_j$ are ways to identify the geometric objects known as tensors. That there are two indices, allowing you to write the components (in some basis) in rows and columns..which then allows you to call it a matrix is not a good reason to do it. I blame computer science for using array and vector to mean the same thing, and intro linear algebra classes, for not distinguishing the abstract number space $\mathbb{R}^3$ from the geometric space ${\bf E}^3$. – JEB Aug 17 '23 at 14:19

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