 Solution

For one-column Gram matrices, unit Euclidean norm of the columns gives $A_S^{\ast }{A}_S=1$. The restricted isometry constant formula therefore gives ${\delta }_1=0$. For a two-column Gram matrix, put $c=⟨a_i,a_j⟩$. The difference from the identity matrix has the formup to the convention for the complex inner product. Its characteristic polynomial is ${\lambda }^2-|c{|}^2$, so the eigenvalues are $\pm |c|$ and its matrix 2-norm is $|c|$. Maximizing over pairs gives ${\delta }_2=\mu$, where $\mu$ is the mutual coherence.

Part (b), together with the minimality defining the restricted isometry constant, gives ${\delta }_s\le {\mu }_1(s-1)$. Every summand defining cumulative coherence is at most the mutual coherence, so ${\mu }_1(s-1)\le (s-1)\mu$. Thus, for normalized columns and $N\ge 2$,The upper range $s\le N$ matters because the printed definition of cumulative coherence stops at $N-1$. For $N=1$, only ${\delta }_1=0$ is needed; the maximum over pairs defining mutual coherence is otherwise empty.