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Orthogonal vector/matrix

Petrus

Well-known member
Feb 21, 2013
739
Hello MHB,
I wounder if I did understand correct, If we got 3 vector and they all are orthogonala, \(\displaystyle v_1=(x_1,y_1,z_1)\),\(\displaystyle v_2=(x_2,y_2,z_2)\),\(\displaystyle v_3=(x_3,y_3,z_3)\) does that also mean that the matrix orthogonal so the invrese for the matrix is transport?

Regards,
\(\displaystyle |\pi\rangle\)
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Close, but not quite. A matrix being orthogonal means that its columns are orthonormal.
 

Petrus

Well-known member
Feb 21, 2013
739
Close, but not quite. A matrix being orthogonal means that its columns are orthonormal.
How can I check if a columns are orthonormal? If I got it correct if \(\displaystyle v_1,v_2,v_3\) shall be orthonormal that means that \(\displaystyle v_1*v_2*v_3=0\)

Regards,
\(\displaystyle |\pi\rangle\)
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Check that $v_{i} \cdot v_{j}=\delta_{ij}$. Here
$$\delta_{ij}=\begin{cases}0,\quad &i \not=j \\ 1,\quad &i=j \end{cases}$$
is the Kronecker delta.