Is orthogonality of column vectors preserved after right-multiplication by unitary matrix? $\mathbf V$ is an $n \times (n-1)$ matrix with mutually orthogonal columns.
$\mathbf Q$ is a unitary matrix of size $(n-1) \times (n-1)$. Is there a concise algebraic proof that the columns of $\mathbf {VQ}$ are mutually orthogonal? 

In a proof of the Spectral Theorem, it is claimed that the columns of $\mathbf {VQ}$ are orthogonal by construction. I would like to find a way to verify that this is generally true.
Let $Q(i)$ be the $i$-th column of $Q$. Think of the product $VQ$ as
$$R = VQ = \begin{bmatrix} VQ(1) &amp; VQ(2) &amp;VQ(3) &amp;\cdots&amp;VQ(n) \end{bmatrix}$$.

We want to show that $\langle R(i), R(j) \rangle = 0,\forall i \neq j$, 
\begin{align*}
&amp;\langle R(i), R(j) \rangle\\
=&amp;Q(i)^TV^T V Q(j) \\
=&amp;Q(i)^TIQ(j)\\
=&amp;Q(i)^TQ(j) \\
=&amp; 0
\end{align*}
Since the columns of $V$ are mutually orthogonal (actually they need to be mutually orthonormal),  $V^TV = I \in R^{n \times n}$. $Q$ is an unitary matrix, therefore its columns are also mutually orthogonal.
A unitary matrix preserves the scalar product, hence preserves the orthogonality of two vectors.