How to find all the non unique solutions in rank deficient system of linear equations in matlab

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Zeab 2020-5-15

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/525590-how-to-find-all-the-non-unique-solutions-in-rank-deficient-system-of-linear-equations-in-matlab

编辑： Zeab 2020-5-19

Dears,

I have a sytem of linear equations and it is rank deficient and therfore the solutions are not unique,how to find all the solutions in matlab?

syms x1 x2 x3 x4 x5;
eqn1 = 2 * x1 -x2 + 2*x3-x4+3*x5 == 14;
eqn2 = x1 + 2*x2+3*x3+x4+0*x5 == 15;
eqn3 = x1 + 0*x2 * -2*x3 +0*x3+ -5 * x5 == -10;
[A, B] = equationsToMatrix ([eqn1, eqn2, eqn3], [x1, x2, x3,x4,x5])

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Answer 1

David Goodmanson 2020-5-15

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/525590-how-to-find-all-the-non-unique-solutions-in-rank-deficient-system-of-linear-equations-in-matlab#answer_432587

编辑：David Goodmanson 2020-5-15

Hello Zeab,

One solution is

v0 = A\b.

Since A is rank defiicient it has a nonempty null space

nullA = null(A)

which is dimension 5x2. Two column vectors span the null space of A, and by definition A*nullA = 0 . The entire solution is v0 plus an arbitrary linear combination of the column vectors in nullA:

syms x1 x2 x3 x4 x5 c1 c2;
eqn1 = 2 * x1 -x2 + 2*x3-x4+3*x5 == 14;
eqn2 = x1 + 2*x2+3*x3+x4+0*x5 == 15;
eqn3 = x1 + 0*x2 * -2*x3 +0*x3+ -5 * x5 == -10;
[A, B] = equationsToMatrix ([eqn1, eqn2, eqn3], [x1, x2, x3,x4,x5])
nullA = null(A)
v0 = A\B
    v0 =
      -10
    -52/7
     93/7
        0
        0
vextra = v0 + nullA*[c1;c2]       % c1,c2 are arbitrary
     vextra =
                      5*c2 - 10
    (29*c2)/7 - (5*c1)/7 - 52/7
        c1/7 - (31*c2)/7 + 93/7
                             c1
                             c2
A*vextra -B  % check
  ans =
    0
    0
    0 