Minimum off-diagonal elements of a matrix with fixed eigenvalues

I have a bound that will be of use to you. First, note that we can use the fact that the diagonal entries are all $1$s to relate $c_\mathrm{max}$ to the Frobenius norm of $C$: $$ \|C\|_F^2\leq M+M(M-1)c_\mathrm{max}^2. $$ This Frobenius norm is easy to work with, since it's just the 2-norm of the spectrum: $$ \|C\|_{F}^2 =\mathrm{Tr}[CC^\mathrm{T}] =\mathrm{Tr}[V\Lambda^2 V^\mathrm{T}] =\mathrm{Tr}[\Lambda^2] =\sum_{m=1}^M\lambda_m^2. $$ Rearranging then produces a lower bound on $c_\mathrm{max}$: $$ c_\mathrm{max}\geq\sqrt{\frac{1}{M(M-1)}\bigg(\sum_{m=1}^M\lambda_m^2-M\bigg)}. $$ Achieving equality in this lower bound certainly implies optimality. For example, consider the following matrix: $$ C =\left[\begin{array}{rrr}1~&-\frac{1}{2}&-\frac{1}{2}\\-\frac{1}{2}&1~&-\frac{1}{2}\\-\frac{1}{2}&-\frac{1}{2}&1~\end{array}\right]. $$ Here, $\Lambda=\mathrm{diag}(\frac{3}{2},\frac{3}{2},0)$, $c_\mathrm{max}=\frac{1}{2}$, and a quick calculation reveals that this achieves equality in our lower bound. But is this always possible?

Unfortunately, no. For example, it's impossible to achieve equality when $\Lambda=\mathrm{diag}(\frac{5}{3},\frac{5}{3},\frac{5}{3},0,0)$. But how do I know that?

Your question is intimately related to another problem that's of use in engineering: Design an ensemble of $M$ unit vectors in $\mathbb{R}^d$, where $M>d$, with the property that no two vectors have a large inner product in magnitude (i.e., you want the ensemble to be incoherent). For this problem, the Gram matrix of the vectors is playing the role of your $C$, and the Welch bound was developed to provide a lower bound on the coherence (your $c_\mathrm{max}$). For details, check out this blog entry.

Your problem has an important distinction from the incoherent design problem: you prescribe the spectrum of $C$. But there's a theorem that says achieving equality in the Welch bound necessitates that the spectrum of $C$ has $\frac{M}{d}$ with multiplicity $d$ and $0$ with multiplicity $M-d$. As such, you might as well consider the instance of your problem in which this is your spectrum (in this instance, the above bound on $c_\mathrm{max}$ is precisely the Welch bound).

The point of looking at this instance is to demonstrate how hard your problem actually is. While there are many Welch-bound achieving ensembles, it is also known that the Welch bound is not always achievable. For example, it is impossible to pack $5$ vectors in $\mathbb{R}^3$ with Welch-bound coherence (this was the source of my second example above, while the first example corresponded to the cube roots of unity in $\mathbb{R}^2$). It's also unknown in general which values of $M$ and $d$ enable Welch-bound equality (in fact, existence of such ensembles is equivalent to the existence of certain strongly regular graphs, and in many cases, existence is a long-standing problem).

For more information about this problem, google "equiangular tight frames" - you just opened a very interesting can of worms. :)

To add some insights into my discussion. I found out some stuff that might be interesting according to matrix perturbation theory.

So instead of asking as in the 1st question: given prescribed eigenvalues what is the minimum $c_{max}$? I tried to look into the problem of what is the possible eigenvalue densities given I have a prescribed $c_{max}$.

It goes like this... I compare my correlation matrix with the identity matrix. The diagonal elements of both matrices are 1's, therefore the perturbation is in the off-diagonal elements. The Weyl-Lidskii theorem states that:

$|\lambda_i-1|\leq ||\mathbf{C}-\mathbf{I}||_2$

where I write $|\lambda_i-1|$ because the eigenvalues of the identity matrix $(\mathbf{I})$ are $1$. But this just leads to some obvious bounds.