Plot graph with complex number

Question

I need some help from expert. I already read about the plot with Identity, I but I not really understand and I think my problem is different and I confuse to used what command of plot.

Here the matrix,

A = {{E^(I*β1 + I*β3)*Cos[β2*t], E^(I*β1 - I*β3)*Sin[β2*t]}, 
     {(-E^((-I)*β1 + I*β3))*Sin[β2*t], E^((-I)*β1 - I*β3)*Cos[β2*t]}}; 
β1 = 0.; 
β2 = Pi; 
β3 = 0.; 
t = 0; 

tu = Table[t++; t = t + TimeUsed[]; Print[A]; t, {i, 1, 10}]

ListPlot[tu]

and the ouput;

{{0.33873792024528937 + 0.*I, -0.9408807689542262 + 0.*I}, 
  {0.9408807689542262 + 0.*I, 0.33873792024528937 + 0.*I}}
{{-0.7705132427757919 + 0.*I, -0.6374239897486864 + 0.*I}, 
  {0.6374239897486864 + 0.*I, -0.7705132427757919 + 0.*I}}
{{-0.8852313113324524 + 0.*I, 0.4651510780774637 + 0.*I}, 
  {-0.4651510780774637 + 0.*I, -0.8852313113324524 + 0.*I}}
{{0.0878511965507509 + 0.*I, 0.9961336091431718 + 0.*I}, 
  {-0.9961336091431718 + 0.*I, 0.0878511965507509 + 0.*I}}
{{0.9529793415172217 + 0.*I, 0.30303526963276495 + 0.*I}, 
  {-0.30303526963276495 + 0.*I, 0.9529793415172217 + 0.*I}}
{{0.6470559615694358 + 0.*I, -0.7624425110114551 + 0.*I}, 
  {0.7624425110114551 + 0.*I, 0.6470559615694358 + 0.*I}}
{{-0.4539904997395583 + 0.*I, -0.891006524188362 + 0.*I}, 
  {0.891006524188362 + 0.*I, -0.4539904997395583 + 0.*I}}
{{-0.9971589002606128 + 0.*I, 0.07532680552794735 + 0.*I}, 
  {-0.07532680552794735 + 0.*I, -0.9971589002606128 + 0.*I}}
{{-0.31498651965528923 + 0.*I, 0.9490961449902997 + 0.*I}, 
  {-0.9490961449902997 + 0.*I, -0.31498651965528923 + 0.*I}}
{{0.7542513807361156 + 0.*I, 0.6565857557529428 + 0.*I}, 
  {-0.6565857557529428 + 0.*I, 0.7542513807361156 + 0.*I}}

ouput of graph;

My problems;

1. I want to plot the result of this matrix but i read that real part cannot be combine with imaginary part, hence How can I plot the graph that I want to consider both during plot (I mean I want to get one coordinate include real and imaginary part). It is possible?
2. I used ListPlot[tu] to represent the data. It is true?

Answer 1

In general, you can always separate the Re and Im part from a data and plot them individually

data = RandomComplex[1 + 1 I, {10, 2}]; (*any set of of complex data*)

data11 = {Re@#[[1]], Re@#[[2]]} &/@ data;
data12 = {Re@#[[1]], Im@#[[2]]} &/@ data;
data21 = {Im@#[[1]], Re@#[[2]]} &/@ data;
data22 = {Im@#[[1]], Im@#[[2]]} &/@ data;

ListLinePlot[{data11, data12, data21, data22}]

Answer 2

This is not a answer, but an extended comment.

Here is your code cleaned up (I reduce the number of iterations from 10 to 3 to save space).

β1 = 0.;
β2 = N[Pi];
β3 = 0.;
a[t_] = 
  {{E^(I*β1 + I*β3)*Cos[β2*t], E^(I*β1 - I*β3)*Sin[β2*t]}, 
   {(-E^((-I)*β1 + I*β3))*Sin[β2*t], E^((-I)*β1 - I*β3)*Cos[β2*t]}} // Chop

{{1. Cos[3.14159 t], 1. Sin[3.14159 t]}, {-1. Sin[3.14159 t], 1. Cos[3.14159 t]}}

Note that there are no complex values.

tu = Module[{t = 0.},Table[t += 1. + TimeUsed[]; Print[a[t]]; i, 3]]

{8.7486, 17.4973, 26.2461}

ListPlot[tu]

The printed values of the matix a[t] are all real-valued, but are not captured by your algorithm. They could had by making a simple modification.

Module[{t = 0.}, Table[t += 1. + TimeUsed[]; a[t], 3]]

{{{-0.74568, 0.666305}, {-0.666305, -0.74568}}, {{0.112157, -0.99369}, {0.99369, 0.112157}}, {{0.578378, 0.815769}, {-0.815769, 0.578378}}}

However, I can not understand from your question how you expect to make a plot of a list of 2 x 2 matrices. Can you make your intent clearer?