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.

Jack Ingram

Jack Ingram

Answered question

2022-10-22

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 A
So 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 = 0
and 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

Answer & Explanation

t5an1izqp

t5an1izqp

Beginner2022-10-23Added 13 answers

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

Valery Cook

Beginner2022-10-24Added 3 answers

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

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