Question

Let `A=((0,1),(0,1))`. Let T linier operator to`R^(2x2)`, and `T(X)=AX-XA`, `AAX``inR^(2x2)`. Determine `rank(T)` ?

Answer 1

We have:

`bb(A) = ( (0, 1), (0, 1) ) \ \ \ `; and; `\ \ ul(bb(T))(bb(X)) = bb(AX) - bb(XA) \ \ AA bb(X) in RR^(2xx2)`

Consider a generic element `bb(X) in RR^(2xx2)`, having terms:

`bb(X) = ( (a_11, a_12), (a_21, a_22) )`

Then consider the effect of the linear operator `ul(bb(T))` on the matrix `bb(X)`;

`ul(bb(T))(bb(X)) = bb(AX) - bb(XA)`

`\ \ \ \ \ \ \ \ \ = ( (0, 1), (0, 1) )( (a_11, a_12), (a_21, a_22) ) - ( (a_11, a_12), (a_21, a_22) )( (0, 1), (0, 1) )`

`\ \ \ \ \ \ \ \ \ = ( (a_21, a_22), (a_21, a_22) ) - ( (0, a_11+a_12), (0, a_21+a_22) )`

`\ \ \ \ \ \ \ \ \ = ( (a_21, a_22-a_11-a_12), (a_21, a_22-a_21-a_22) )`

`\ \ \ \ \ \ \ \ \ = ( (a_21, a_22-a_11-a_12), (a_21, -a_21) )`

The vectors:

`( (a_21), (a_21) ) \ \ \ `; and; `( (a_22-a_11-a_12), (-a_21) )`

Are linearly independent, and therefore:

`rank(ul(bb(T))) = 2`

even though `rank(bb(A))=1`