# Prove that a basis is orthonormal if and only if its matrix is unitary

If $$V=C^n$$ (Complex numbers) unitary, inner product vector space such that $$dim_CV=n$$ and $$\beta=$${$$u_1,u_2,...,u_n$$} basis. Prove that $$\beta$$ is orthonormal if and only if the matrix

so $$A=\begin{pmatrix}\leftrightarrow&A_1 &\leftrightarrow\\\ \cdots & \cdots& \cdots\\\leftrightarrow&A_n& \leftrightarrow\end{pmatrix}=\begin{pmatrix}\leftrightarrow&u_1 &\leftrightarrow\\\ \cdots & \cdots& \cdots\\\leftrightarrow&u_n& \leftrightarrow\end{pmatrix} \in M^{nxn}(C)$$

is unitary, i.e, A is inversible and $$A^{-1}=A^*$$

I would be glad if I can get some help, I don't even know where to start, I know that a basis is orthonormal if $$=0$$ $$\forall i\neq j$$ and if $$||u_k||=1$$ $$\forall k$$

• Yes I do, I had no idea how to represent that in a better way in Latex – Juju9704 Jun 12 at 12:03
• Just note that $(AA^{*})_{ij}=\langle u_i,u_j\rangle$ – Peter Melech Jun 12 at 12:06

Hint: Consider what each entry of $$AA^*$$ is, and while you do, look hard at your definition of orthonormal basis.
• I think that for $AA^*$, the entries are $\delta_{ij}$, right? – Juju9704 Jun 12 at 12:07