How to show an arbitrary set is orthonormal?

Question

I have to show that the rows and columns of a unitary matrix are orthonormal sets. I have not been provided with a specific matrix, just 'a unitary matrix.' I understand how to show that given vectors are orthonormal, and I know that if the rows of a unitary matrix are orthonormal, then the columns are as well, but I can't think of how to prove this with an arbitrary matrix. Any help is appreciated.

Answer 1

Two (complex) vectors are orthogonal iff $$\sum_i a_i^*b_i = 0$$ A matrix $U$ is unitary iff $U^\dagger U = I$.

A set of vectors is orthonormal if each vector in the set has norm of $1$, that is, $\sum a_i^*a_i = 1$, and for all pairs of two distinct vectors in the set, $\sum_i a_i^*b_i = 0$.

Suppose, now, that $U$ is unitary, and let $\vec{a}$ be the $k$-th column of $U$, and let $\vec{b}$ be the $j$-th column of $U$, with $k\neq j$. Then by the rules of matrix multiplication $$\sum_i a_i^*b_i = (U^\dagger U)_{kj} = I_{kj} == 0 $$ so $\vec{a}$ and $\vec{b}$ are orthogonal.

Now let $\vec{b}$ be the $k$-th column of $U$ so that the two columns are the same column. Then $$\sum_i a_i^*a_i = (U^\dagger U)_{kk} = I_{kk} == 1 $$ so $\vec{a}$ is of norm $1$, that is to say, it is normal.

Thus the set of vectors is orthonormal.