sympy矩阵未对齐

Question

I'm trying to use sympy to help me isolate a vector in a matrix expression. I have written this code:

    import sympy
    from sympy import symbols, MatrixSymbol, Matrix
    from sympy import Identity    

    Xp = MatrixSymbol('Xp', 3,1)
    t = MatrixSymbol('t', 3,1)
    n = MatrixSymbol('n', 3,1)
    H = n.T*Xp*Identity(3) - t*n.T

my intention is to isolate n. I'm not sure if sympy can do that, but I already get a 'ShapeError: Matrices n'*Xp and I are not aligned', I think this error should not happen as n'*Xp is a scalar so it should be able to multiply with a matrix.

How can I get the expression nT*Xp*Identity(3) to be valid? and, can sympy help me to isolate the vector n in this equation?

Answer 1

n.T*Xp*Identity(3) 

has the dimension signature

(1,3)*(3,1)*(3,3)

which obviously will not work.

n*Xp.T*Identity(3) 

could work.

---

If you want to solve

H=(n.T*X)*I-t*n.T

for n then the first remark is that this is not always possible. Assuming that a solution exists, remark that

1/(t.T*t)*t.T*H=1/(t.T*t)*(n.T*X)*t.T-n.T

so that

n = a*t - b*H.T*t

where a is unknown and b=1/(tT*t) . Inserting into the original equation gives

H = (a*t.T*X-b*t.T*H*X)*I - a*t*t.T + b*t*t.T*H

or

H - b*t*t.T*H +b*(t.T*H*X)*I = a*((t.T*X)*I - t*t.T)

which in each non-trivial component of the right side matrix will give a value of a , but a solution only exists if all those values are the same.