Invert a matrix from a 4D array : equivalence or difference with indexes

[fab13]

TL;DR

I am looking for help about the manipulation in a formula of the inversion of matrixes from a 4D array and the interpretation of a factor which is actually a matrix of 100x100 size in this formula.

I have a 4D array of dimension $100\text{x}100\text{x}3\text{x}3$. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions $100\text{x}100$ for the following equation (it allows to compute the $(i,j)$ element $F_{ij}$ of Fisher matrix) :

$F_{ij} = \sum_{XY}\,\int\,\dfrac{V_{0}\,\text{d}^3 k}{(2\pi)^{3}}\,\bigg(\dfrac{\partial P_{\text{obs,X}}}{\partial p_{i}}\bigg)\,(C^{-1})_{XY}\,\bigg(\dfrac{\partial P_{\text{obs,Y}}}{\partial p_{j}}\bigg)\quad(1)$

$X$ and $Y$ vary between $1,2,3$.
The derivatives $\bigg(\dfrac{\partial P_{\text{obs,X}}}{\partial p_{i}}\bigg)$ and $\bigg(\dfrac{\partial P_{\text{obs,X}}}{\partial p_{j}}\bigg)$ are discrete arrays of dimension $100\text{x}100$.

First I compute a "Covariance matrix" $C$ of dimension $100\text{x}100$ for each $X$ and $Y$. So I have a 4D array for $C$.

1) Now, I have an issue about the interpretation of the term $(C^{-1})_{XY}$.

For the moment, what I do is to compute a 2D array of matrix $3\text{x}3$ (so a 2D array of $10000$ elements with a matrix $3\text{x}3$ for each element of this 2D array).

Then, after this, I invert the $10000$ matrixes $3\text{x}3$ and I multiply by a direct dot product of $C^{-1}$ with the derivatives in parenthesis.

To compute th final value $F_{ij}$, I do a summation on $X$ and $Y$ (i.e on $X,Y=1,2,3$) of the 2D final grid $100\text{x}100$ and integrate it.

2) I don't know if things of this formula are well carried out. Indeed, inverting $10000$ matrixes $3\text{x}3$ is not the same thing than inverting 9 matrixes of size $100\text{x}100$, is it ?

In all cases, I thing I have to consider the matrix $(C^{-1})_{XY}$ like a matrix of $100\text{x}100$ dimensions.

Indeed, I think $(C^{-1})$ should be a $100\text{x}100$ matrix.

But on the other side, in my code, I perform the inversion of $10000$ $3\text{x}3$ matrixes and once this computation is done, I take the 9 matrixes of $100\text{x}100$ dimension.

Is my method right ?

3) I suppose that the 2 following operations are not commutative :

invert $10000$ matrixes of $3\text{x}3$ and using 9 matrixes of $100\text{x}100$ from this inversion OR invert directly 9 matrixes of $100\text{x}100$

It seems that I make confusions between inverting matrix and reconstruct or use again this inversion with a 4D array.

If someone could help me to interpret correctly the equation $(1)$, especially the term $(C^{-1})_{XY}$, in order to do the right computations.

Regards

 

[tnich]

No, in general inverting 10,000 3x3 matrices will not give you the same result as inverting 9 100x100 matrices. If you want to convince yourself of that, try it both ways with a 2x2x3x3 array and see if you get the same result.

 

[fab13]

Sorry, I add more details since my first post may cause confusions.

I did a little error in the equation $(1)$ on notations of Covariance factor $C$, the right one is :

$F_{ij} = \sum_{XY}\,\int\,\dfrac{V_{0}\,\text{d}^3 k}{(2\pi)^{3}}\,\bigg(\dfrac{\partial P_{\text{obs,X}}}{\partial p_{i}}\bigg)\,C_{XY}^{-1}\,\bigg(\dfrac{\partial P_{\text{obs,Y}}}{\partial p_{j}}\bigg)\quad(1)$

What I am almost sure is that factor $C$ must have a size of $100\text{x}100$.

I think I have 2 choices for the factor $C_{XY}^{-1}$ in the equation $(1)$ if I use a 4D dimension array for $C_{XY}^{-1}[0:100][0:100][0:2][0:2]$ , i.e with an element noted by $C_{XY}^{-1}[\text{i}][\text{j}][\text{k}][\text{l}]$ :

1) Invert for each $(i,j)$ element a matrix $3\text{x}3$, then I take each block of $100\text{x}100$ associated to each element of matrix $3\text{x}3$ , and after multiplying each derivatives block (of size $100\text{x}100$) by each of the $(k,l)$ elements (which varies from $0$ to $2$) and which represent a block $100\text{x}100$ . In this case, the formula would mean a dot-to-dot product of each element $(i,j)$ of the matrix $C_{XY}^{-1}$ (of dimensions $100\text{x}100$), by the corresponding element $(\text{i},\text{j})$ of $X,Y$ derivatives blocks (also of dimensions $100\text{x}100$)

2) Invert for each $(k,l)$ element a matrix of size $100\text{x}100$ and multiplying with a dot product each of the 9 matrixes $100\text{x}100$ by each of the corresponding derivatives (labeled by index $X$ and $Y$).

For the moment, a colleague has implemented the first solution 1) but I have not enough elements to confirm if that is the right method.

I have a few difficulties to explain better the 2 approaches because there is a slightly difference but results between both will never be the same, at least I think.

QUESTION 1) : What I assume also checkable is that operation 1) and 2) don't give the same $100\text{x}100$ matrix for the factor of inverted covariance $C_{XY}^{-1}$ : is it right ?

QUESTION 2) : What do you think about these 2 possible interpretations to compute $C_{XY}^{-1}$, which one to use ?

Any help would be great because I am stuck for a long time on this issue. Regards