Representing a matrix as a scalar Is there a way to represent a matrix as a scalar without losing information? I know that I can use something as the Frobenius Norm, but that does not account for change in element positions. For example:
The F norm of the (1x3) matrix (2, 1, 0) is equal to the F norm of a matrix like (1, 2, 0). But the matrices are different. So is there a way to represent a matrix by a scalar and also detect these position changes?
UPDATE:
My practical problem here is I have a measurement process that outputs two matrices for a specific measurement. Both are square, symmetrical and integer matrices.
First one looks like:
     [,1] [,2] [,3] [,4]
[1,]    0    1    1    0
[2,]    1    0    0    1
[3,]    1    0    0    0
[4,]    0    1    0    0

Second one:
     [,1] [,2] [,3] [,4]
[1,]    0    0    1    0
[2,]    0    0    0    1
[3,]    1    0    0    0
[4,]    0    1    0    0

Now, suppose I have a chain of measurements and need to detect an "anomalous" measurement (say, the matrix elements change radically in position and/or value). What I need to face this problem is to somehow convert a matrix to a single value and then perform anomaly detection on this chain of scalars. I thought about using the norm, but the posiotining information is lost, like I described above.
 A: $\mathbb{R}^{n\times m}$ (real $n\times m$ matrices) and $\mathbb{R}$ (the real line) are bijective as sets, but you don't even need that much. You just need a unique real number for each unique matrix, so an injective function. Bijectivity implies injectivity, so such a function must exist. (Whether or not it is practical is another issue.)
For your binary matrices, such a function is easy to construct. Let's go through an example.
$$\begin{pmatrix}a & c\\\ b & d\end{pmatrix}$$
Since the values are all $0$ or $1$, we can represent the matrix just as $a + 10b + 100c + 1000d$, and this gives straightforward unwrapping like:
$$\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix} \rightarrow 1110$$
If other words, you just unwrap the matrix and place the digits next to each other.
If you don't have constant dimensions, you can unwrap the matrix and put a $2$ after each column. The above example would be:
$$\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix} \rightarrow 1212120$$
Perhaps try to figure out the matrix corresponding to $2121212120$.
You may have more complicated matrices than this, but I think this example shows that you can write an injective function that maps a set of matrices to  scalar values.
(You have, however, asked about an XY problem, where you have issue X (matrix anomalies) and have idea Y (matrices as scalars) to solve X, and you've asked about Y instead of X. Your real question about matrix anomalies would be a great fit on Cross Validated where you originally posted this question. That is a statistical question, and a pretty good one, too. You may want to consider posting about that on Cross Validated.)
A: (Note) The question was moved, but I think the OP found this answer useful. If elaboration or examples are desired then a supplied example form of matrix would be helpful.
You are trying to take many pieces of information and boil them down to a single numeric representation.
This is a good early question. Greats in the field engaged this question starting hundreds of years ago. There are still good things to consider here. Consider generative, before a decade ago it wouldn’t have been on the menu. Good answers here relate to physics and mathematics in many significant and important ways.
There are ways to do this.

*

*One could use a generative tool in machine learning to convert a scalar to a matrix. That tells you part of the problem though: it depends on what you’re trying to do with it.

*you could take a mean, arithmetic, geometric, harmonic, or other.

*You could compute the determinant

*You could compute a Norm, there are an infinite number of families of norms many of which have infinite members

*You could compute a statistical summary, such as the maximum, the minimum, a Quantile or interquantile, the variance or other moments, or one of infinite other statistics

*you could take the largest eigenvalue, or any statistical summary on the eigenvalues

*many other options

If you presume that it is a rotation, you could scale it, and then use geometric methods to extract the minimum angle of the rotation in a hyper sphere.
And this is the problem, the place were magic happens, because what you do to it has implicit assumptions. Without a clean articulation of what those assumptions are, it is impossible to get to the particular numeric value that is useful to you.
