I'm trying to show that a matrix A is normal iff the dot product of the i,j row vectors of A = dot product of j,i column vectors of A.
Assuming A is normal, I used the equation AA*=A*A and rewrote it so that i,jth element is
\[\sum\limits_{k=1}^n a_{ik}\overline{a}_{jk}=\sum\limits_{k=1}^n \overline{a}_{ki}a_{kj}\]
but it seems that we can rewrite the LHS as
\[\sum\limits_{k=1}^n a_{ik}\overline{a}_{jk}=\langle a_i,a_j\rangle\] and the RHS as
\[\sum\limits_{k=1}^n \overline{a}_{ki}a_{kj}=\overline{\langle c_i,c_j\rangle}=\langle c_j,c_i\rangle.\] where cj and ci are the column vectors of A.
Then I was just going to go backwards and conclude that A is normal.
First of all, I'm not sure that this is notationally correct... Am I on the right track?
Assuming A is normal, I used the equation AA*=A*A and rewrote it so that i,jth element is
\[\sum\limits_{k=1}^n a_{ik}\overline{a}_{jk}=\sum\limits_{k=1}^n \overline{a}_{ki}a_{kj}\]
but it seems that we can rewrite the LHS as
\[\sum\limits_{k=1}^n a_{ik}\overline{a}_{jk}=\langle a_i,a_j\rangle\] and the RHS as
\[\sum\limits_{k=1}^n \overline{a}_{ki}a_{kj}=\overline{\langle c_i,c_j\rangle}=\langle c_j,c_i\rangle.\] where cj and ci are the column vectors of A.
Then I was just going to go backwards and conclude that A is normal.
First of all, I'm not sure that this is notationally correct... Am I on the right track?