I was reading through a research paper on compressive sensing, and I came across the following:
Let $z \in d_{z}$ denote a vectorized image with $d_{z}$ pixels. Assume that $z$ is $K$–sparse in a basis given by the columns of $\Psi \in \Re^{d_{z} \times d_{z}}$, meaning that $z = Ψθ$ and at most $K$ of the coordinates of $θ$ are nonzero. Compressive sensing theory ensures that $\theta$, and therefore $z$, can with high probability be exactly recovered from appropriate linear projections onto the rows of a measurement matrix $\phi \in \Re^{d_{y}×d_{z}}$ , with $d_{y} < d_{z}$. Specifically, define the $coherence$ $\mu$ between $\phi$ and $\psi$ as $\mu \equiv \sqrt{d_{z}} \max_{i,j} \langle\phi_{i}, \psi_{j}\rangle \in [1, \sqrt{d_{z}}]$ for all rows $\phi_{i}$ and columns $\psi_{j}$ of $\phi$ and $\psi$, respectively;
What is the meaning of the equation $μ ≡ \sqrt{d_{z}} \max_{i,j} <\phi_{i}, \psi_{j}> \in [1, \sqrt{d_{z}}]$ in the above context?
Also, is $\mu$ a scalar value or a vector?
Construct the matrix whose ij entry is $\phi_i \cdot \psi_j$. Then take the maximum entry of that matrix. This gives a scalar. Multiplying by the other scalar $\sqrt{d_z}$ gives another scalar for $\mu$. The angle brackets for the dot product seems to have been the source of your confusion.