How to find unknown linear operator?
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I have a square matrix A that acts on a some vector x such that A*x = y. Matrix A is heavily constrained with only a few unknowns. x and y are known exactly. How do I solve for A?
For example say:
sym a b c d
A = [a 0 0 b;
c d 0 0;
0 0 0 0;
0 0 0 0]
x = % some known vector
y = % some known vector
How do we find A now?
2 个评论
sahil kommalapati
2019-11-23
编辑：sahil kommalapati
2019-11-23
This is a method which come to mind:
syms a b c d
A = [a 0 0 b;
c d 0 0;
0 0 0 0;
0 0 0 0];
x = zeros(4,1); %your known vecotor here
y = zeros(4,1); %and here!
eq1 = A*x ==y;
k = solve(eq1, [a,b,c,d]);
sol = [k.a, k.b, k.c, k.d]
David Goodmanson
2019-11-23
Hello Tsuyoshi,
Well, with your example for A, you are certainly not going to do it for arbitrary x and y, because the form of A means that y(3)=y(4)=0. So any solution with either y(3) or y(4) nonzero is impossible. (It's a consequence of A being singular).
But let's say that y(3)=y(4)=0. The two equations that are left are
a*x(1) + b*x(4) = y(1)
c*x(1) + d*x(2) = y(2)
which is one eqn. for two unknowns a,b
and one eqn. for two unknowns c,d
so you can get solutions, but you can't solve for any unknown uniquely.
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