Contents

A.2 Generate example matrix suitable for row-reduction

My most-used example generator is the genrowred function which starts with a random matrix in REF, then does \(n + m\) random row operations.

Warning

Most \(m\times n\) matrices created with integers suffer from the problem that row-reducing them by-hand often produces ugly denominators.

The genrowred function itself calls several other functions, all of which are shown below.

Tip

The subroutines shown below use the sample command from the Statistics and Machine Learning Toolbox and also require the Symbolic Math Toolbox.

The genrowred function code

function myMatrix = genrowred(M, N, k)
    % Input Validation
    arguments
        M (1,1) {mustBeInteger, mustBePositive}
        N (1,1) {mustBeInteger, mustBePositive}
        k (1,1) double = M + N;
    end
    % Additionally, n <= 8
    m = min(8, M);
    n = min(8, N);

% Initialize matrix C to ensure the while loop runs at least once
C = eye(2)*100*n*m;

while (sum(sum(C))) > max(100,10 * m * n ) | (sum(sum(isnan(C))) > 0)
    A = genpivots(m,n);
    B = fillAbovePivots(A);
    C = fillZeroCols(B);
    D = randRowOps(C,k);
end
myMatrix = D;
% Can use code below to add result to clipboard
clipboard('copy',myMatrix);
end

Watch it run

Note

\(A\) has pivots only.

\(B\): integers above pivots.

\(C\): integers in zero-columns.

An REF matrix \(C\) results.

I executed the code block below and printed out the final matrices \(A,B,C,D\) the function produced.

genrowred(3,5)
\[\begin{split}A = \left[\begin{array}{rrrrr}4&0&0&0&0\\0&4&0&0&0\\0&0&0&3&0\\\end{array}\right] \\ B = \left[\begin{array}{rrrrr}4&5&0&0&0\\0&4&0&-1&0\\0&0&0&3&0\\\end{array}\right] \\ C = \left[\begin{array}{rrrrr}4&5&0&3&0\\0&4&5&-1&0\\0&0&0&3&5\\\end{array}\right] \end{split}\]

Note

\(D\) is the result of \(k=5\) random row operations on \(C\). User can set \(k\).

\[\begin{split}D = \left[\begin{array}{rrrrr}-40&-42&10&-17&25\\40&46&-5&16&-25\\4&5&0&3&0\\\end{array}\right]\end{split}\]

The Subroutines

genpivots

function myMatrix = genpivots(M,N)
% Input Validation
    arguments
        M (1,1) {mustBeInteger, mustBePositive};
        N (1,1) {mustBeInteger, mustBePositive};
    end
    % Additionally, m,n <= 8
    m = min(10, M);
    n = min(10, N);

% Determine maximal square dimensions inside matrix
minLength = min(m,n);

% Random diagonal square matrix with dimension minLength
A = gendiag(minLength);

% Add rows or columns of zeros until matrix M has dimensions m x n
while ( length(A) < m )  | ( length(A) < n )
    if length(A) < m
        v = zeros(1,length(A(1,:)));
        A = [A;v];
    else
        v = zeros(length(A(:,1)),1);
        A = [A , v];
    end
end

    % Take zero-columns from end, inserts them randomly in between non-zero-columns.
    if m < n
        [m,n] = size(A);
        p = 1:length(A);
        q = p;
        r = max(m,n);
        s = min(m,n);
        t = r-s;
        v = sort(randsample(1:s,min(t,s)), 'descend');
        for k = v
            p = [p(1:k),p(end),p(k+1:end-1)];
        end
    A(:,q) = A(:,p);
    end

% Move all zero-rows to bottom of matrix, required for REF
myMatrix = zRows2btm(A);
end

Function zRows2btm

function myMatrix = zRows2btm(A)
[m,n] =size(A);

b = [];
t = [];

for i = 1:m
    if sum(A(i,:)) == 0
        b = [b,i];
    else
        t = [t,i];
    end
end
[t b];
A([1:m],:) = A([t b],:);
myMatrix = A;

fillAbovePivots

function myMatrix = fillAbovePivots(A)
    arguments
        A (:,:,:)
    end

[m,n] = size(A);

for j = 1:n
        nZ = firstNonZero(A(:,j));
        if ( nZ <= m) & ( nZ > 1 ) & ( sum(A(:,j) ~= 0) )
            for i = 1:(nZ - 1)
                A(i,j) = randsample([-2:5],1);
            end
        end
    end
myMatrix = A;
end

fillZeroCols

function myMatrix = fillZeroCols(A)
    % Input Validation
    arguments
        A (:,:,:)
    end
[m,n] = size(A);
for j = 2:n
    for i = 1:m
        if ( A(i,j) == 0) & ( sum(A(i,1:j)) ~= 0 )
            A(i,j) = randsample(-2:5,1);
    end
end
    
myMatrix = A;
% Can use code below to add result to clipboard
% clipboard('copy',myMatrix);
end

randRowOps

RowOp1: Swap rows

RowOp2: Non-zero scalar \(\times\) row

RowOp3: Replace row with of itself and scalar multiple of another row

randRowOp1

function myMatrix = randRowOp1(M)
[m,n] = size(M);
r = randsample(1:m,2);
p = r(1);
q = r(2);
M([p q],:) = M([q p],:);
myMatrix = M;
end

randRowOp2

function myMatrix = randRowOp2(M)
[m,n] = size(M);
k = randsample(1:m,1);
r = randsample(2:5,1);
M(k,:) = M(k,:) * r;
myMatrix = M;
end

randRowOp3

function myMatrix = randRowOp3(M)
[m,n] = size(M);
k = randsample(1:m,2);
p = k(1);
q = k(2);
r = randsample([-2:-1 2:3],1);
M(p,:) = M(p,:) + M(q,:)* r;
myMatrix = M;
end

function myMatrix = randRowOps(A,N)
    % Input Validation
    arguments
        A (:,:,:) 
        N (1,1) {mustBeInteger, mustBePositive}
    end
    % Additionally, n <= 8
    n = min(20, N);

[m,q] = size(A);
    
for i = 1:n
    k = randsample(1:6,1);
    if k == 1
        A = randRowOp1(A);
    else if k < 3
            A = randRowOp2(A);
        else
            A = randRowOp3(A);
        end
    end
end

for i = 1:m
    A(i,:) = A(i,:) / gcd(sym( A(i,:)));
end

myMatrix = A;
end
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