Question 1   What conditions must the vectors $\{x_i\}_{i=1}^k \subset {\mathbb R}^n$ satisfy in order to be a tight frame in ${\mathbb R}^n$

Solution 1   Let $E$ be the $k \times n$ matrix whose rows are the vectors $\{x_i\}_{i=1}^k$ $$ E = \begin{pmatrix} x_{i,1} & \cdots & x_{i, n} \\ \vdots & \ddots & \vdots \\ x_{k,1} & \cdots & x_{k,n} \end{pmatrix}. $$ Then the tight frame condition $\sum_{i=1}^k | \langle x, x_i \rangle |^2 = A \|x\|^2 $ gives $(Ex)^TEx=Ax^Tx$ for all $x \in {\mathbb R}^n$ or $E^TE = AI_n$ $$ E^TE = \begin{pmatrix} \sum_{i=1}^k x_{i,1} x_{i,1} & \cdots & \sum_{i=1}^k x_{i,1} x_{i,k} \\ \vdots & \ddots & \vdots \\ \sum_{i=1}^k x_{i,k} x_{i, 1} & \cdots & \sum_{i=1}^k x_{i,k} x_{i, k} \end{pmatrix} = \begin{pmatrix} A & \cdots 0 \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots 0 \cdots & A \end{pmatrix} = AI_n. $$

Therefore, the vectors $$ \left\{ x_i = \begin{pmatrix} x_{i,1} \\ \vdots \\ x_{i,n} \end{pmatrix} \right\}_{i=1}^k \subset {\mathbb R}^n $$ are an A-tight frame iff the vectors $$ \left\{ x^\prime_i = \begin{pmatrix} x_{1,i} \\ \vdots \\ x_{k,i} \end{pmatrix} \right\}_{i=1}^n \subset {\mathbb R}^k,$$ i.e., the columns of $E$ are all of norm $\sqrt{A}$ and form an orthogonal family.