If $P$ is an $n\times{n}$ orthogonal matrix prove that the columns of $P$ form an orthonormal set in $R^n$. So I know that a matrix is orthogonal if $AA^T$=$I$ but I'm not sure how that would help me here. Any help is appreciated.
Linear Algebra – Orthogonal Matrix Columns Forming an Orthonormal Set
linear algebramatrices
Best Answer
If $A A^{\top} = I$, then $A^{\top} = A^{-1}$, so that $A^{\top} A = I$.
Now note that the $(i, j)$-entry of $A^{\top} A$ is the scalar products of the $i$-th row of $A^{\top}$ (that is, the $i$-th column of $A$) by the $j$-th column of $A$.
Note the general fact that a single matrix equality compactly and conveniently encodes several scalar identities.