Hello,
I have the following system of equation Ax = B where A is a 2x4 matrix, B is a 2x1 and x is a 4x1 matrix.
A = [ a b 0 0]
0 0 c d]
I am looking for X can I use a pseudo inverse matrix here , if no? may you tell me how to proceed or what other method to use.
Thank you in advance .
Hi, Khalil.
Unlike your previous system, this one is underdetermined, so it has infinite number of solutions [z1,z2,z3,z4], where (z1,z2) is any point of the line a*x1+b*x2=B1 and (z3,z4) is any point of the line c*x3+d*x4=B2. Your B=[B1,B2]T.
Best regards,
Victor
Dear Khalil,
You Linear system represent 2 independent straight lines.
The solutions are determined by direct solution:
The general solution is in the attached file.
Best regards
Hi, Khalil!
Yes, you can use Moore–Penrose inverse (i.e. pseudoinverse) to get the solution... But everything depends on what do you want to get. In case of over-determined linear system pseudo-inversion will lead to the best solution in sense of least squares; in case of under-determined linear system (this is your case) it will lead to the shortest solution (in sense of euclidean norm) among all of possible solutions.
As Isaam and Victor noted, your system has infinite number of solutions; So, if you need to get the shortest solution - use pseudo-inverse.
See, example in Python (using numpy):
# Solve ax = b using pseudoinverse...
>>> import numpy as np
>>> a = np.array([[1,2.4, 0, 0], [0,0,4.5,6]]) # lets define
>>> np.linalg.pinv(a)
array([[0.14792899, 0. ],
[0.35502959, 0. ],
[0. , 0.08 ],
[0. , 0.10666667]])
>>> b=np.array([5,6]) # lets define
>>> b=np.array([5,6.7])
>>> b
array([5. , 6.7])
>>> x = np.linalg.pinv(a)@b # @ stands for matrix production
array([0.73964497, 1.77514793, 0.536 , 0.71466667]) # x -- the shortest solution
Using Dmitry's idea and Issam's solutin we can easily find the unique shotest solution:
[a b1/(a2+b2), b b1/(a2+b2), c b2/(c2+d2), d b2/(c2+d2)]
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