Importing a large number of complex matrices from a .DAT file and then numerically analyzing them one by one

Question

I have a .DAT file generated by Fortran in which I have data for $2 \times 2$ matrices. For example, the file takes the form (when I open it in the excel):

I have added a snippet beacause the real variables are too long to type in TeX font. As you can see, each row contains 4 terms, corresponding to an entry in the matrix. Each bracket contains 2 terms because each entry is defined as a complex number; we have a complex matrix at hand.

Now there are about 1500 such entries in the file. Each entry correspond to a distinct time for which the matrix has been calculated. I want to do the following and I am not sure how to do the following steps:

For a specific time, say $t = 0$ (first entry), get the $2 \times 2$ matrix. Do some manipulations on them. The manipulations are of form: multiplying the matrix with 2 other matrices (twice) to get 2 scalar functions (one from each set of multiplication); taking the difference of the 2 scalar functions; and numerically optimizing the difference and get the optimized values of the inputs. I then want to repeat the analysis for all the other entries (matrices) in the file.

How does one go about doing such analysis in Mathematica? I'm clueless on this front. I know how to import one real matrix. I don't know how to import one complex matrix; and how to recursively import matrices from different rows and do the analysis on each matrix seperately.

If I know how to approach the problem, I can tackle it and then post additional questions with codes and related problems.

P.S: Please do suggest any other appropriate tag, if any.

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Edit:

Using

Import["FILEPATH", "Table"]

here are the first few entries in the imported list:

{{( 0.70992637972979133     ,  0.0000000000000000     ), 
 ( 0.34768150998932623     ,-0.29148928933886220     ), 
  ( 0.34768150998932623     , 0.29148928933886220     ), 
   ( 0.29007362027020861     ,  0.0000000000000000     )},        {0.71578138101774014     ,  0.0000000000000000     ), (
 0.34162400932308312     ,-0.29395228624439607     ), (
    0.34162400932308312     , 0.29395228624439607     ), (
        0.28421861898225981     ,  0.0000000000000000     )}}

Fullform is:

 List[List["( 0.70992637972979133     ,  0.0000000000000000     )","(   0.347"\[Ellipsis] "0     )",""\[Ellipsis] "","( 0.29007362027020861     ,  0.0000000000000000     )"],\[LeftSkeleton]1500\[RightSkeleton]]

Answer 1

You are reading Strings. So please review if Import option settings are correct. There is a whole range of formats for tabular, database and spreadsheet files. If you are sure this is as good as it gets:

input=Import["FILEPATH", "Table"];

I would first split the Strings at the round brackets:

firstpass=StringSplit[input, "(" | ")"];

Then replace the text strings with actual expressions (values). You should now have a list structure with values (check this with FullForm):

secondpass = ToExpression[firstpass];

This may be what you need, i.e. a list of 2x2 matrices; but if dimensionality is still not correct, do this:

thirdpass=Partition[Flatten[secondpass],{2,2}]

This will restructure your flat list into a series of 2x2 matrices. You may need to fiddle this a bit as I don't have the file so could not test it on your data.

So let's assume you get your list of 2x2 matrices and want to manipulate these. First I create a random matrix list with the same format:

fourthpass = RandomReal[{-1, 1}, {1000, 2, 2}];

UPDATE: The rest is speculation, as you do not specifiy what you want to do, however this may be applicable for manipulating several matrices, e.g. matrix multiplication of two subsequent matrices:

FoldList[#1 #2 &, IdentityMatrix[2], fourthpass]//Rest;

or perhaps raising the entire list to the third power

MatrixPower[#,3]&/@fourthpass;

or

myownoptimizationfunction[#]&/@fourthpass;

In general, you do not need loop constructs. The above is more straigtforward and often faster.