So I have a matrix vector space consisting of the three matrices: $$ e_1 = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1 \end{pmatrix} e_2 = \frac{1}{\sqrt{6}} \begin{pmatrix} 1 & 1 & 1\\ 0 & 0 & 0\\ -1 & -1 & -1 \end{pmatrix} e_3 = \frac{1}{\sqrt{6}} \begin{pmatrix} 1 & 0 & -1\\ 1 & 0 & -1\\ 1 & 0 & -1 \end{pmatrix} $$ I am suppose to show that these matrices are orthonormal in the vector space. How is this to be achieved? I know that the requirements are that each base matrix {$e_1, e_2, e_3$} need to have the norm 1 and be pairwise orthogonal, i.e. $e_i \cdot e_j = 0$ for all $i \neq j$ in the vector space.
I can calculate the matrix norm as the maximum absolute column sum, hence $||e_1|| = |\frac{1}{3} + \frac{1}{3} + \frac{1}{3}| = 1$. However the norm $||e_2|| = |\frac{1}{\sqrt{6}} - \frac{1}{\sqrt{6}}| = 0$. Therefore $e_2$ does not fulfill the requirements for an orthonormal basis. Am I doing anything wrong here?
Also, how am I suppose to calculate the pairwise ortogonal basis matrices? I.e. how am I suppose to calculate the scalar product of two matrix e.g. $e_1 \cdot e_2$? How does one calculate the scalar product of two matrices?
According to my professor these basis matrices are suppose to give an orthonormal vector space hence my confusion.