proof of property

Jun 2019
13
0
London
Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.
 

romsek

Math Team
Sep 2015
2,773
1,552
USA
do you mean $A^TA=0$

Consider the 3x3 identity matrix.

Clearly all the column vectors are orthogonal, and $Iv = v$

$I^T=I$

$I+I= 2I \neq 0$
 
Oct 2018
126
94
USA
I may have read this wrong, but I don't believe $I_{3 \times 3}$ can be $A$ since it would mean $Iv$ is orthogonal to $v$.
 
Jun 2019
13
0
London
I may have read this wrong, but I don't believe $I_{3 \times 3}$ can be $A$ since it would mean $Iv$ is orthogonal to $v$.
Yeah, A is definitely not an identity matrix. I think that maybe A is zero matrix. Or maybe I have forgotten some theorem or criteria.
 
Last edited by a moderator:
Oct 2018
126
94
USA
This was somewhat long so I wouldn't be surprised if there's an error in here somewhere so read critically.

Let $A = \begin{bmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{bmatrix}$

and $v= \begin{bmatrix}
v_1 \\
v_2\\
v_3
\end{bmatrix}$

Two vectors are othogonal if their dot product is $0$, so we know $Av \cdot v = 0$

$Av = \begin{bmatrix}
a_1 v_1 + a_2 v_2 + a_3 v_3 \\
b_1 v_1 + b_2 v_2 + b_3 v_3 \\
c_1 v_1 + c_2 v_2 + c_3 v_3
\end{bmatrix}$

$Av \cdot v = v_1 (a_1 v_1 + a_2 v_2 + a_3 v_3)\\
+ v_2 (b_1 v_1 + b_2 v_2 + b_3 v_3)\\
+ v_3 (c_1 v_1 + c_2 v_2 + c_3 v_3) = 0$

after a bit of simplifying:

$Av \cdot v = a_1 v_{1}^{2} + b_2 v_{2}^{2} + c_3 v_{3}^{2} + v_1 v_2 (a_2 + b_1) + v_1 v_3 (a_3 + c_1) + v_2 v_3 (b_3 + c_2) = 0$

So , if $a_1 , b_2 , c_3 = 0$ and $b_1 = -a_2 $ , $c_1 = -a_3$ , and $c_2 = -b_3$ then $Av \cdot v = 0$

So $A=\begin{bmatrix}
0 & a_2 & a_3 \\
-a_2 & 0 & b_3 \\
-a_3 & -b_3 & 0
\end{bmatrix}$

And you then have $A^{t} + A = 0$
 
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Oct 2018
126
94
USA
$Av \cdot v = a_1 v_{1}^{2} + b_2 v_{2}^{2} + c_3 v_{3}^{2} + v_1 v_2 (a_2 + b_1) + v_1 v_3 (a_3 + c_1) + v_2 v_3 (b_3 + c_2) = 0$
If $a_1 , b_2 , c_3 = 0$ , then $a_1 v_{1}^{2} + b_2 v_{2}^{2} + c_3 v_{3}^{2} =0$ for any $v$