characterization of tight frames in ${ℝ}^n$

Question 1.

What conditions must the vectors ${\mathrm{\{}{x}_i\mathrm{\}}}_{i\mathrm{=}\mathrm{1}}^k\mathrm{\subset }{\mathrm{R}}^n$ satisfy in order to be a tight frame in ${\mathrm{R}}^n$?

Solution.

Let $E$ be the $k\times n$ matrix whose rows are the vectors ${\{x_i\}}_{i=1}^k$:

Then the tight frame condition ${\sum }_{i=1}^k{|⟨x,x_i⟩|}^2=A{\parallel x\parallel }^2$ gives ${(Ex)}^{T}Ex=A{x}^{T}x$ for all $x\in {ℝ}^n$, or $E^{T}E=A{I}_n$:

Therefore, the vectors

$${\{x_i=\left(\begin{array}{c}\hfill x_{i,1}\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_{i,n}\hfill \end{array}\right)\}}_{i=1}^k\subset {ℝ}^n$$

are an A-tight frame iff the vectors

$${\{x_i^{\prime }=\left(\begin{array}{c}\hfill x_{1,i}\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_{k,i}\hfill \end{array}\right)\}}_{i=1}^n\subset {ℝ}^k,$$

i.e., the columns of $E$, are all of norm $\sqrt{A}$ and form an orthogonal family.