So I have a matrix
$$
A =
\frac{1}{3}
\begin{pmatrix}
0 & 1 & 0\\
1 & 1 & 1\\
1 & 0 & 1\\
1 & 1 & 1
\end{pmatrix}
$$
that I am suppose to show as orthonormal.
I know that the conditions for an orthonormal are that the matrix must have vectors that are pairwise independent, i.e. their scalar product is 0, and that each vector's length needs to be 1, i.e. ||v|| = 0.
However I don't see how this can apply to the matrix A? Let $v_1$ be the leftmost vector and $v_2$ be the middle vector then $v_1 \cdot v_2 \neq 0$ and $||v_1|| \neq 1$ hence A can't be orthonormal.
In my assignment I am suppose to show that this matrix is orthonormal, however, according to the defintion A is not orthonormal. Am I missing something here?
Best Answer
The concept of othonormal matrix is defined only for square matrices, and $A$ is not one such matrix. And, yes, those columns are not orthogonal.