Here is a table.  Is it a matrix?

Is it meaningful too...

• ... reorder along an row/column?
• ...rescale along a row/column?
• ...add one row to another, or do so with columns?

Bus tables are not matrices

"matrix" elevates data to a math;

thus, it must comply with rules.

A table which cannot be explored linearly is not a matrix.

Is a Grade Book a matrix?

Are graphs matrices?

Adjacency: $A_{ij}$ if $(i,j)$ is an edge.

Node-arc: rows=nodes, columns=edges,

+1 if start of edge, -1 if end of edge.

Are graphs matrices?

No, you cannot do linear algebra on the rows and columns without destroying the graph.

...and yet

• If $A$ is the matrix then $A^n$ records the paths of length $n$
• If $A$ is a node-arc matrix of a graph, then row/column elimination is essential for linear programming, max flows, min-cuts, etc..
• Spectral graph theory is another deep subject.

These are linear connection; but require nuance.

Is an image a matrix (2-tensor)?

• It is a rows and columns with scalar entries
• Folks do linear algebra on images (I was told in class years ago)

...and yet...

• If $M$ is an image matrix then the eigenvectors of $M^* M$ tell us close approximations to the image.  (Singular value decomposition more generally).
• Shifting $M$ to a phase space with a cosine transform lets us compress the image close to faithfully.

Once more, linear algebra is lurking, but you have to work for it.

Is a video a 3-tensor?

• It is a 3-axis grid of scalars.
• Folks do linear algebra with videos, but its not directly on the axes.

Is a neural network a tensor?

Certainly not across the non-linear part.

Linear algebra is scalpel not a hammer.

If you attack a grid as a tensor,

you better show reason to expect the results to be meaningful.

Yes, but...

(the rebuttal)

If your advisor, boss, or mentor calls every array a "tensor" then by all means go with it.

(You might just call actual tensors "formal tensors")