Vector, u := [ u 1 , … , u n ] T . I am trying to find a coordinate transformation matrix, Q ∈ R n × n , which is nonsingular, satisfying: [ 0 ⋮ 0 | | u | | ] = Q u .

Question

Vector,   u  :=  [      u    1    ,  …  ,      u    n        ]          T      . I am trying to find a coordinate transformation matrix,   Q  ∈            R              n      ×      n      , which is nonsingular, satisfying:                              [                                                    0                                                                    ⋮                                                                    0                                                                                      |                                                  |                                u                                  |                                                  |                                                              ]                =        Q        u        .

Abagail Stephenson · Accepted Answer

Recall that if   W is a subespace of             R              n      , then  dim  ⁡  (  W  )  +  dim  ⁡  (      W          ⊥        )  =  nTake   W as the span of   u, so  dim  ⁡  (  &amp;lt;  u  &amp;gt;  )  =  n  −  1.You can construct a matrix   Q that will satisfy the same condition taking any set of   n  −  1 vectors       v    i   from any basis of the ortogonal complement of the span of   u.  Q  =      (                                        v                          11                                                            v                          12                                                ⋯                                      v                          1              n                                                                        v                          21                                                            v                          22                                                ⋯                                      v                          2              n                                                                        v                          31                                                            v                          32                                                ⋯                                      v                          3              n                                                            ⋮                          ⋮                          ⋮                          ⋮                                                  v                          n              −              1              ,              1                                                            v                          n              −              1              ,              2                                                ⋯                                      v                          n              −              1              ,              n                                                                                                                        u                  1                                                  |                  |                                      u                                    |                  |                                                                                                                                          u                  2                                                  |                  |                                      u                                    |                  |                                                                              ⋯                                                                                      u                  n                                                  |                  |                                      u                                    |                  |                                                                          )  where       v          i        =  (      v          i      1        ,  …  ,      v          i      n            )          T       are vectors in a basis   {      v          1        ,  …  ,      v          n      −      1        } of the ortogonal complement of the linear span of   г, for all   i with   1  ≤  i  ≤  n  −  1.

Darius Miles · Accepted Answer

This matrix   Q is non singular because it&#039;s rows are linearly independent because the following set is a basis of             R              n      :      B    =  {  u  ,      v    1    ,  …  ,      v          n      −      1        }Also: You can use Gram Schmidt process to get an ortogonal basis of   &amp;lt;  u      &amp;gt;          ⊥       starting from   u. In this way you will find the rights vectors       v    i   such that   Q is inversible.

Vector, u:=[u_1,…,u_n]T^. Find a coordinate transformation matrix, Q in R^(nxn), which is nonsingular, satisfying: [0,...,0,||u||]=Qu

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