(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p
I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that CT=[I_p 0] where "I_p" is identity matrix of dimension "p". As I have studied the transformation matrix "T" will exist if rank(C)=p.
For Eg. Suppose C=[1 0 2;2 3 1] of dimension 2 by 3, here p=2 and n=3 and rank(C)=p=2, then how will we find matrix "T" of dimension "3 by 3" such that CT=[1 0 0;0 1 0]=[I_2 0].
I require the above transformation matrix "T" in control problem for state transformation.
(2.) From my point of view
T can be written as
T=C^{+}[I_p 0]+Y[I_n-C^{+}C]
where "Y" is any arbitrary matrix of dimension "n by n". C^{+} is right pseudo inverse of C and equivalent to C^{+}=C'(CC')^{-1} where C' is transpose of C.
So if one multiply both sides of above equation with C then get required solution CT=[I_p 0] because CC^{+}=I_p, so above solution gives different values of "T" for different values of "Y" and thus a general solution.
(3.) If I take Y=0 i.e. null matrix then rank(T)is less then n, which I don't want. If I set Y=I_3 then get the required transformation matrix.
(4.) I have studied there is some similarity transformation exist to do above task but I don't have much knowledge about it.