I have a matrix in which each number represents a physical property (spin) of an atom. The whole matrix represents the state of my system. for example :
x={{0,1,2},{-1,-2,-3},{-1,0,2}}//MatrixForm
or
y={{0,2,2},{-1,-2,-3},{-1,0,2}}//MatrixForm
represent two different (among 20 000 other states) and therefore orthogonal states.
now I have several states which are linear combinations of my basis states for example
state1=ax+by
state2=a'x+b'y.
I want to orthogonalize these state via the gram schmidt process (or something like that).
Q:Is there a way to tell mathematica that two different matrices are orthogonal in this special case (i.e. that x and y are othogonal)?
or in other words is there a way to make mathematica perform the orthogonalization process without having to rewrite the basis states into a series of 1 an 0 ?
I was thinking of using the f option in the documentation of the Orthogonalize function but I can't see how. (I am kinda new to mathematica)
Thank you very much for your help.