How can I obtain the generic matrix T such that X ^ = T X with: X ^ = [ a 0 0 b c 0 0 d ] and X = [ a b c d ] I did several try without obtaining the result I needed.

Question

How can I obtain the generic matrix T such that             X      ^        =  T  X with:            X      ^        =      [                            a                          0                                      0                          b                                      c                          0                                      0                          d                      ]  and  X  =      [                            a                          b                                      c                          d                      ]  I did several try without obtaining the result I needed.

Illuddybopylu · Accepted Answer

Is X is invertible? Then just multilpy       X          −      1       at right of both side of equation.Then T will be             X      ^            X          −      1

Staffangz · Accepted Answer

There is no such T that&#039;s independent of the entries of X.If there were such T, then consider the effect of swapping the columns of X,      X    ′    =      [                            b                          a                                      d                          c                      ]    =  X      [                            0                          1                                      1                          0                      ]  On one hand, T was assumed to be generic and would transform X′ to the same format:  T      X    ′    =      [                            b                          0                                      0                          a                                      d                          0                                      0                          c                      ]  On the other hand, TX′ should be like TX but with swapped columns:  T      X    ′    =  T  X      [                            0                          1                                      1                          0                      ]    =      [                            a                          0                                      0                          b                                      c                          0                                      0                          d                      ]        [                            0                          1                                      1                          0                      ]    =      [                            0                          a                                      b                          0                                      0                          c                                      d                          0                      ]  And note how the zero entries are at different positions.

How can I obtain the generic matrix T such that hat(X)=TX with: hat(X)=[[a,0],[c,0],[0,b],[0,d]] and X=[[a,c],[b,d]]

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