What are there different rules you need to check for with an orthogonal matrix or vector? Why does an orthogonal matrix need to nxn but a orthogonal vector does not?
$$\begin{bmatrix} 1\\ 0\\ 1\end{bmatrix}\begin{bmatrix} 0\\ 1\\ 0\end{bmatrix}$$
dot product 1*0 + 0 * 1 + 1 *0 = 0
What else do I need to check for besides dot product with a matrix? Why does this have to be nxn?
\begin{bmatrix} 1 & 0\\ 0 & 1\\ 1 & 0 \end{bmatrix}
dot product 1*0 + 0 * 1 + 1 *0 = 0
What else do I need to check for besides dot product with a vector?
What about one like this? Why is this not orthogonal?
\begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix}
dot product 1*1 + 1 * -1 = 0
$\endgroup$ 81 Answer
$\begingroup$A matrix whose columns are orthonormal vectors is called an Orthogonal matrix. If A is orthogonal, you can verify by multiplication that ATA is identity, where AT is A transposed. This is true even if A is rectangular. But AAT is not identity. If A is square then A*AT is also identity and then Ainverse=AT
$\endgroup$ 1
