I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative question is easier to answer.
First way
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Assume I have a real symmetric matrix $\mathbf{C} \in \mathbb{R}^{M \times M}$.
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I know its eigenvalues that are non-negative: $\lambda_1, \ldots, \lambda_M$. And The trace of the matrix, i.e., the sum of all eigenvalues is $\lambda_1+\cdots+\lambda_M = M$.
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The diagonal matrix of eigenvalues is $\mathbf{\Lambda}$ and the matrix with eigenvectors in its colums is $\mathbf{V}$. The eigendecomposition is then $\mathbf{C} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^T$.
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Also the diagonal of the matrix is all ones, i.e., $\operatorname{diag}(\mathbf{C}) = [1,\ldots,1]$.
Define $$c_\max = \max\limits_{i\neq j}|c_{ij}|$$ where $c_{ij}$ is the element of $\mathbf{C}$ in the $i$-th row and $j$-th column. Given that I can choose $\mathbf{V}$ freely, i.e., any matrix with those eigenvalues, what is the minimum of the maximum of all off-diagonal elements that I can attain (in absolute value)? In other words what is the minimum of $c_\max$?
Second way
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Given that you have $M$ vectors $\{\mathbf{v}_1, \ldots, \mathbf{v}_M\}$.
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They are orthonormal $\mathbf{v}_i^T \mathbf{v}_j = \delta(i-j)$ by standard dot product definition.
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They have norm one $\| \mathbf{v}_i \|=1$ by standard dot product definition.
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Define the weighted inner product as $\mathbf{v}_i^T \mathbf{\Lambda} \mathbf{v}_j$, where $\mathbf{\Lambda} = \operatorname{Diag}(\lambda_1,\ldots,\lambda_M)$ and $\operatorname{trace}(\mathbf{\Lambda}) = M$.
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$\{\mathbf{v}_1,\ldots,\mathbf{v}_M\}$ also have norm one $\|\mathbf{v}_i\|_w=1$ by this new weighted inner product definition.
What is then the minimum value for the maximum inner product (in absolute value) among all vectors $\{\mathbf{v}_1,\ldots,\mathbf{v}_M\}$ given they can be chosen freely as far as they satisfy the conditions?
$$\min\limits_{\mathbf{v}_1,\ldots,\mathbf{v}_M} \left(\max\limits_{i\neq j}(\mathbf{v}_i^T\mathbf{\Lambda}\mathbf{v}_j)\right)$$
Best Answer
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. :)