Creating Matrix provided elements

2021 7 22
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更新时间：2021 7 24
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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?

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Image Analyst
Image Analyst 2021-7-23
Did you experiment around with the real(), imag(), eye(), tril(), and triu() functions? If not, why not? That's what I'd do.
Chunru
Chunru 2021-7-23
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
Jan 2021-7-23
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
Karthik Nagaraj 2021-7-23
编辑：Karthik Nagaraj 2021-7-23
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

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Jan
Jan 2021-7-24
编辑：Jan 2021-7-24
Ran in:

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Ran in:
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

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Karthik Nagaraj
Karthik Nagaraj 2021-7-24
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!

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