Creating Matrix provided elements

From a Hermititan (complex skew symmetric) matrix of order N (Asssume N=15) a column vector is created such that all the diagonal elements are placed first and then the ordered pair of real and imaginary parts of upper triangle matrix are placed next. Since it is hermitian matrix the upper and lower triangle elements have same set of real and imaginary elements.

For example for N=15x15 matrix the vector looks like this

[D1, D2, D3,...........,D15, R11, I11,R12, I12,.... ,R15, I15] in total 225 elements column vector.

How to construct back the matrix given this vector?

[Image Analyst]

Did you experiment around with the real(), imag(), eye(), tril(), and triu() functions? If not, why not? That's what I'd do.

[Chunru]

For Hermitian matrix, the following are independent variables:

[D1, D2, ... D15, R1_2, I1_2, ..., R1_15, I1_15, R2_3, I2_3, ... R2_15, I2_15, ....]

Try to get the input corrrect before next step.

[Jan]

What is R1_2 compared to L1_2? Should it be L2_1? If it is a hermitian matrix, why are the L elements stored?

Please explain exactly, what the inputs are. Use a 4x4 matrix to avoid the need to use unclear abbreviations.

[Karthik Nagaraj]

Hermitian Matrix N=4. Always Diagonal are real. Upper triangle elements are complex conjuagte of Lower triangle elements

disp(H)
   0.9490 + 0.0000i  -0.0222 + 0.8156i  -1.2365 + 0.4256i  -0.1310 - 0.4833i
  -0.0222 - 0.8156i   0.3080 + 0.0000i  -0.0918 + 1.2658i  -0.3776 - 0.1952i
  -1.2365 - 0.4256i  -0.0918 - 1.2658i  -0.9261 + 0.0000i  -0.0571 - 0.9267i
  -0.1310 + 0.4833i  -0.3776 + 0.1952i  -0.0571 + 0.9267i   0.2858 + 0.0000i
  

I have created a funciton that can extract Diagonal elements, Upper triangle elements in ordered pair (R-Real and I-Imaginary element one after the other and form a vector of these.

Diagonal elements

disp(D)
    0.9490
    0.3080
   -0.9261
    0.2858

Upper triangle elements ordered pair. Real and imaginary pair next to each other

disp(UOP)
   -0.0222    0.8156   -1.2365    0.4256   -0.0918    1.2658   -0.1310   -0.4833   -0.3776   -0.1952   -0.0571   -0.9267
   

The final concatenated vector VU.

  Columns 1 through 13

    0.9490    0.3080   -0.9261    0.2858   -0.0222    0.8156   -1.2365    0.4256   -0.0918    1.2658   -0.1310   -0.4833   -0.3776

  Columns 14 through 16

   -0.1952   -0.0571   -0.9267

I have more than 1000 vectors like 'VU'. I need to reconstruct the oroginal matrix H with vector VU given.

Some similar Matlab forum solution is given in

https://www.mathworks.com/matlabcentral/answers/350413-put-elements-into-corresponding-locations-of-upper-triangular-matrix#answer_275680

But could not improvise further

Answer 1

[Jan]

A = rand(4) + 1i * rand(4);
A = A + A';   % Hermitian

% Convert to vector:
D  = diag(A).';
L  = triu(A, 1);
Lf = L(L ~= 0).';
Lv = [real(Lf); imag(Lf)];
VU = [D, Lv(:).'];

% And backwards:
n  = sqrt(numel(VU));
L  = triu(ones(n), 1);
L(L==1) = VU(n+1:2:n*n) + 1i * VU(n+2:2:n*n);
% Or:  L(L==1) = [1, 1i] * reshape(VU(n+1:n*n), 2, [])
B  = diag(VU(1:n)) + L + L';

isequal(A, B)

ans = logical
   1

[Karthik Nagaraj]

The first part of converting matrix to vector looks exactly as my function. But the code for reconstruction of the matrix back from the vector is really good. This must be one of the best answers! Thank you!