Firstly, the definition of RIP constant is given as
Let $\mathbf{A}$ be an $m\times p$ matrix and let $1 \leq s \leq p$ be an integer. Suppose that there exists a constant $\delta _{s}\in (0,1)$ such that every $s$-sparse vector $\mathbf{v}$ satisfies
\begin{align} (1-\delta _{s})||\mathbf{v}||_2^2 \leq ||\mathbf{A}\mathbf{v}||_2^2 \leq (1+\delta _{s})||\mathbf{v}||_2^2 \end{align}
Then, the matrix $\mathbf{A}$ is said to satisfy the $s$-restricted isometry property with restricted isometry constant $\delta_s$.
My question: Does the matrix $\mathbf{A}$ need to be column normalized? If not column normalized, the eigen values of random generated $\mathbf{A}^H\mathbf{A}$ could be very large. For example, when the eigenvalues are $[5 \;4.9 \;4.8\; 4.7]$, the columns of matrix $\mathbf{A}$ are actually quasi-orthognal. However, it does not satisfy the RIP property defined above.
I've searched over many literatures, but almost all of the highly cited work neglect the column normalization.
Please refer to Elad's book, where the columns need to be $l_{2}$ normalised (chapter 5 page 86).