Determining Smith Form of a rectangular matrix

Question

Akshay Vivek Panchwagh 2022-7-5

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https://ww2.mathworks.cn/matlabcentral/answers/1754105-determining-smith-form-of-a-rectangular-matrix

评论： Akshay Vivek Panchwagh 2022-7-7

I need to find Smith Form of a rectangular 3x2 matrix in MATLAB. I searched for documentation but all the solutions it has are for square matrices. Can anyone please help me in determining how I can find the Smith Form using smithForm for a rectangular matrix?

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Answer 1

Rohit Kulkarni 2022-7-5

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https://ww2.mathworks.cn/matlabcentral/answers/1754105-determining-smith-form-of-a-rectangular-matrix#answer_1000895

Hi,

I think this may help you https://in.mathworks.com/matlabcentral/fileexchange/60410-mnsmithform-a.

Open the Functions tab, which is besides the overview tab, it contains the function to evaluate smith form.

I don't think there is any way to use "smithForm" to find the smith form of non-square matrices and one will have to write an algorithm to compute it.

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Rohit Kulkarni 2022-7-7

在 MATLAB Online 中打开

Hey, I tried some examples myself, I think its working right

A = [40 -20 36 12; -24 -4 28 48; 0 4 0 -8; 8 0 0 -8]

A = 4×4

40 -20 36 12 -24 -4 28 48 0 4 0 -8 8 0 0 -8

MNsmithForm(A)

ans =

B = [1 1; -2 6; 0 8]

B = 3×2

1 1 -2 6 0 8

MNsmithForm(B)

ans =

C = [6 -6 4; -6 -12 -8]

C = 2×3

6 -6 4 -6 -12 -8

MNsmithForm(C)

ans =

Function in the link provided in the answer above:

function [SA, invFact, D] = MNsmithForm(A)
row = size(A,1);
col = size(A,2);
n = min(row,col);
minors = cell(1,n);
D0 = 1;
D0 = sym(D0);
D = sym(NaN(1,n));
invFact = sym(NaN(1,n));
for i = 1:n;
    rowindex = false(1,row);
    rowindex(1:i) = true;
    rowperms = unique(perms(rowindex),'rows');
    colindex = false(1,col);
    colindex(1:i) = true;
    colperms = unique(perms(colindex),'rows');
    rownum = size(rowperms,1);
    colnum = size(colperms,1);
    minors{i} = sym(NaN(rownum,colnum));
    for j=1:rownum;
        for k=1:colnum;
            Atmp = A;
            Atmp = Atmp(rowperms(j,:),:);
            Atmp = Atmp(:,colperms(k,:));
            minors{i}(j,k) = det(Atmp);
        end
    end
    rowlen = rownum*colnum; %(row - (i-1))*(col - (i-1));
    minors{i} = reshape(minors{i},1,rowlen);
    minors{i}(minors{i} == 0) = [];
    D(i) = gcd(minors{i});
    if i == 1
        invFact(i) = D(i)/D0;
    else
        invFact(i) = D(i)/D(i-1);
    end   
end
SA = diag(invFact);
if row>col
    zerorows = zeros(row-col,col);
    SA = [SA;zerorows];
elseif col>row
    zerocols = zeros(row,col-row);
    SA = [SA, zerocols];
end
end