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There is an aspect of tensor-index notation that I want to know more about. For a simple example, let [itex]M[/itex] be a [itex]1[/itex]-[itex]1[/itex]-tensor (ie., a matrix), whose elements can be indexed by Latin letters.

What is the difference between the component [itex]M^a_{\hspace{2mm}b}[/itex] and the component [itex]M_b^{\hspace{2mm}a}[/itex]?

The fact that there is a difference between the two components must indicate that the [itex]M[/itex] in the latter case is not the same matrix as the [itex]M[/itex] in the former case. Otherwise, since in both cases [itex]M[/itex] is a rank-[itex]2[/itex] tensor with one contravariant component, indexed above by [itex]a[/itex], and one covariant component, indexed above by [itex]b[/itex], the component [itex]M^a_{\hspace{2mm}b}[/itex] and the component [itex]M_b^{\hspace{2mm}a}[/itex] are both referring to the same matrix element, namely, that in row [itex]a[/itex] and column [itex]b[/itex]. So, if in both cases [itex]M[/itex] is the same matrix, then they both refer to the same matrix element of the same matrix, and therefore must be the same.

So, if [itex]M^a_{\hspace{2mm}b}[/itex] refers to a component of the matrix [itex]M[/itex], then [itex]M_b^{\hspace{2mm}a}[/itex] must refer to a component in the same row and same column of a different matrix, which I will call [itex]M^\#[/itex], just as [itex]M_a^{\hspace{2mm}b}[/itex] doesn't refer to a component of [itex]M[/itex], but instead refers to a component of [itex]M^T[/itex].

Firstly, could someone please confirm that this is what is going on?

More importantly, is there a name for the transformation that changes the matrix with components [itex]M^a_{\hspace{2mm}b}[/itex] to that with components [itex]M_b^{\hspace{2mm}a}[/itex]? In other words, is there a name for what I used the symbol [itex]\#[/itex] for in the previous paragraph, analogous to the transpose operation [itex]T[/itex] at the end of the paragraph?

Moving beyond the simple case of a [itex]1[/itex]-[itex]1[/itex]-tensor, is there a general name for operations which take an [itex]m[/itex]-[itex]n[/itex]-tensor indexed by [itex]m+n[/itex] indices, and switches some of the indices horizontally only (ie., switches around the location of contravariant and covariant indices, but doesn't make any contravariant components covariant or vice versa)?

Finally, does anyone have a good

*interpretation*of the difference between the matrices [itex]M[/itex] and [itex]M^\#[/itex], ie., an intuitive sense of what it mathematically means to switch the horizontal position of two indices of a matrix (and, more generally, for a tensor)? Perhaps a very simple example would be rotations - if [itex]M[/itex] represents a rotation about some axis (not necessarily in Euclidean space, if a different space is needed for the index switch to matter), what exactly does [itex]M^\#[/itex] represent?

Thanks very much for any help that you can give.

-HJ Farnsworth