The Orthogonality of Vectors and Matrices

[Petrus]

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]

Close, but not quite. A matrix being orthogonal means that its columns are orthonormal.

 

[Petrus]

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]

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.

 

[blue_raver22]

Hello |\pi\rangle,

Yes, you are correct. If three vectors are orthogonal, it means that they are perpendicular to each other. This also means that they are linearly independent, which leads to an orthogonal matrix. An orthogonal matrix is a square matrix where all the columns are orthogonal to each other, and the inverse of an orthogonal matrix is equal to its transpose. This is because the transpose of a matrix represents its rotation, and an orthogonal matrix represents a rotation in a 3D space. Therefore, the inverse of an orthogonal matrix is the opposite rotation, which is achieved by transposing the matrix.

I hope this helps clarify your understanding of orthogonal vectors and matrices.

Best regards,