[Math] Orthogonal matrix and orthonormal columns

linear algebramatrices

Show that the columns of orthogonal matrix are always orthonormal.
Hint: $A^TA=I$

Can't really get even started, I thought that it has to be orthonormal since the result is $I$ and not only a diagonal matrix with other numbers in them?

Best Answer

Writing matrix A = $\begin{bmatrix} {v_1} & {v_2} ... & {v_n} \\ \end{bmatrix}$ as built from column vectors and making computation $A^TA$ (which we know is equal $I$) gives result:

$\begin{bmatrix} {v_1}^T\\ {v_2}^T\\ ... \\ {v_n}^T\\ \end{bmatrix}\begin{bmatrix} {v_1} & {v_2} & ...& {v_n}\\ \end{bmatrix}=\begin{bmatrix} {v_1}^T{v_1} & {v_1}^T{v_2} & ...& {v_1}^T{v_n}\\ {v_2}^T{v_1} & {v_2}^T{v_2}& ...&{v_2}^T{v_n}\\ ... & .... & ... &.... \\ {v_n}^T{v_1}& {v_n}^T{v_2} & ... & {v_n}^T{v_n}\\ \end{bmatrix}$ =$\begin{bmatrix} 1 & 0 & ... &0\\ 0 & 1 & ... &0\\ ... & .... & ...&.... \\ 0 & 0 & ...& 1\\ \end{bmatrix}$

what gives wanted interpretation.

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