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 .
You can use the Moore-Penrose Inverse P of A (e.g. in sage: P=A.pseudoinverse() ). Then x = PB is the unique solution such that the norm of Ax - B is minimal, and under all these the norm of x is minimal.
Khalil
For a generic real or complex m x n matrix A,
There is an n x m matrix G such that GA = I exactly when rank(A) = n (which forces n =< m) exactly when Ax= b has exactly zero or one solution for each choice of b in R^m or C^m.
There is an m x n matrix H such that AH = I exactly when rank(A) = m (which forces n >= m) exactly when Ax = b has at least one solution for each choice of b in R^m or C^m.
More generally, when rank(A) < min{m,n}, neither G nor H as used above exists. Nonetheless, there are always n x m matrices G such that GAG = G and AGA = A (this implies rank(G) = rank(A)). If you impose the two most commonly used additional constraints on G, you will obtain the unique generalized inverse called the Moore-Penrose pseudo-inverse, often denoted by either A^+ or A with a superscript dagger symbol. One common way to produce this particular generalized inverse is to produce a rank revealing factorization for A, construct the Moore-Penrose pseudo-inverse of each of the factors, and then take the reverse product of the factor pseudo-inverse.
For your particular matrix A, which is 2 x 4, the rank is zero, one or two depending on the values of a,b,c,d. In all cases the Moore-Penrose pseudo-inverse will be 4 x 2. If all four of these parameters are zero, then the Moore-Penrose pseudo-inverse must be the 2 x 4 all zeros matrix.
As another example, suppose that a and c are both nonzero, so rank(A) is two. Let g = a^2 + b^2 and let h = c^2 + d^2. Then the Moore-Penrose pseudo-inverse G is givenn by
[a/g 0
b/g 0
0 c/h
0 d/h ].
Notice that GA = I (the 2 x 2 identity) in agreement with the fact that rank(A)= m = 2.
One can build the Moore-Penrose pseudo-inverse for all of the different combinations of zero versus nonzero a,b,c,d .
I hope that this is helpful.
Jeff Stuart
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