3x3 Real Matrix -B and A
A-1BA=B' and B-1AB=A'
Diagonalization of B, A
|* Diagonalization of the matrix: applicable for non-singular matrices. A-1BA=B' or BA=AB' where B' is a diagonal matrix whose diagonal elements are λ1 ,λ2, λ3.|
| B= b1 b2 b3 and A = a1 a2 a3 then [
b1 -λ1 b2 b3
[ b1 -λ2 b2 b3
[ b1 -λ3 b2 b3
b4 b5 b6 a4 a5 a6 b4 b5-λ1 b6 a4 =0 ; b4 b5-λ2 b6 a5 =0 b4 b5-λ3 b6 a6 = 0
b7 b8 b9 a7 a8 a9 b7 b8 b9-λ1] a7 ] b7 b8 b9-λ2] a8 ] b7 b8 b9-λ3] a9 ]
* The general characteristic equation is [ b1 -λ b2 b3 * [x
b4 b5-λ b6 y =0 where there are 3 solutions to λ and for each λ, (x,y,z) are (a1,a4,a7), (a2,a5,a8) & (a3,a6,a9) respectively .
b7 b8 b9-λ] z ]
* since these are homogeneous equations , the matrix is singular and hence out of x,y,z one has to specify value of either x,y,z and then others can be found out.