The matrix A has size 3 × 3 and we know that for any column vector v ∈ R 3 the vectors A v and v are orthogonal. Prove that A T + A = 0, where A T is the transposed matrix ASo if A = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) and v = ( x y z ) orthogonality of Av and v brought me to the equation x ⋅ y ⋅ ( a 12 + a 21 ) + x ⋅ z ⋅ ( a 13 + a 31 ) + y ⋅ z ⋅ ( a 23 + a 32 ) + a 11 ⋅ x 2 + a 22 ⋅ y 22 + a 33 ⋅ z 2 = 0and the matrix A T + A equals A T + A = ( 2 a 11 a 12 + a 21 a 13 + a 31 a 12 + a 21 2 a 22 a 23 + a 32 a 13 + a 31 a 23 + a 32 2 a 33 ) However I don't see how then prove that A T + A = 0

Question

The matrix A has size   3  ×  3 and we know that for any column vector   v  ∈            R              3       the vectors   A  v and v are orthogonal. Prove that       A          T        +  A  =  0, where       A          T       is the transposed matrix   ASo if  A  =      (                                        a                          11                                                            a                          12                                                            a                          13                                                                        a                          21                                                            a                          22                                                            a                          23                                                                        a                          31                                                            a                          32                                                            a                          33                                            )  and  v  =      (                            x                                      y                                      z                      )  orthogonality of Av and v brought me to the equation  x  ⋅  y  ⋅  (      a          12        +      a          21        )  +  x  ⋅  z  ⋅  (      a          13        +      a          31        )  +  y  ⋅  z  ⋅  (      a          23        +      a          32        )  +      a          11        ⋅      x          2        +      a          22        ⋅      y          22        +      a          33        ⋅      z          2        =  0and the matrix       A          T        +  A equals      A          T        +  A  =      (                            2                      a                          11                                                            a                          12                                +                      a                          21                                                            a                          13                                +                      a                          31                                                                        a                          12                                +                      a                          21                                                2                      a                          22                                                            a                          23                                +                      a                          32                                                                        a                          13                                +                      a                          31                                                            a                          23                                +                      a                          32                                                2                      a                          33                                            )  However I don&#039;t see how then prove that       A          T        +  A  =  0

t5an1izqp · Accepted Answer

It suffices to show   ⟨  (      A    ⊺    +  A  )  v  ,  v  ⟩  =  0 for all   v. This is true since                    ⟨        (                  A          ⊺                +        A        )        v        ,        v        ⟩                            =        ⟨                  A          ⊺                v        ,        v        ⟩        +        ⟨        A        v        ,        v        ⟩                                          =        ⟨        v        ,        A        v        ⟩        +        ⟨        A        v        ,        v        ⟩                                          =        2        ⋅        ⟨        A        v        ,        v        ⟩                                          =        2        ⋅        0                                          =        0.            Remark: Note that this holds for all real square matrices A.

Valery Cook · Accepted Answer

Since all vectors v are orthogonal to Av then       v    T    A  v  =  0 and likewise   (  A  v      )    T    v  =      v    T        A    T    v  =  0 and so adding them together we get       v    T    A  v  +      v    T        A    T    v  =      v    T    (  A  +      A    T    )  v  =  0 and finally since v was arbitrary we must have   A  +      A    T    =  0

The matrix A has size 3 xx 3 and we know that for any column vector v in RR^3 the vectors Av and v are orthogonal. Prove that A^T+A=0, where A^T is the transposed matrix A.

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