Finding the eigenvalues of a matrix with very regular block structure

Question

I have a matrix $W$ that can be described by the following block structure

$$W=\begin{pmatrix} 0_n & sC\\ sC^T & f(I_m-J_m) \end{pmatrix}$$

where

• $J_m$ is an $m \times m$ matrix only made up of ones

• $0_n$ is the $n \times n$ zero matrix

• $I_m$ is the $m \times m$ identity matrix

• $C$ is an $n \times m$ matrix whose entries are $\pm 1$

• $f > 0$ and $s > 0$ are real numbers.

So, this matrix has a highly regular structure. Most importantly, it is symmetric, so I know it must have $n+m$ real eigenvalues and eigenvectors. Are there any results, that could help me find the eigenvalues (and hopefully also the eigenvectors) of this kind of matrix?

If I directly try to compute the characteristic polynomial, this gets very messy, even though this matrix has this regular structure. The best I can do is simplify the expression

$$\det(W-\lambda I_{n+m})=\det\begin{pmatrix} -\lambda I_n & sC\\ sC^T & (f-\lambda)I_m-fJ_m) \end{pmatrix}$$

using the equivalence

$$\det\begin{pmatrix} A & B\\ C & D\end{pmatrix} = \det(A) \det(D-CA^{-1}B)$$

if $A$ is invertible, to write

$$\det(\lambda f(I_m-J_m)-\lambda^2I_m-s^2C^TC)=0$$

as an equation for the non-zero eigenvalues of $W$. But I fail to simplify this any further.

EDIT:

The reason I am interested in this, is to understand quadratic forms like $x^T\; W\; D\;x$, where $D$ is some positive diagonal matrix (or even $0\leq d_i\leq 1$. I am mostly interested in upper or lower bounds of these expressions.

Answer 1

This doesn't provide an answer but was way too long for a comment and contains an extended example that can give relevant insights in the problem so I decided to post it anyway.

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Let us consider the simplest case of $m=n=1$. Out matrix then becomes $$\begin{pmatrix}0&sC\\ sC^T &0\end{pmatrix}.$$ Note that $C=\pm 1$ We may then calculate the characteristic polynomial to $$p(\lambda)=\lambda^2-s^2C^TC=\lambda^2-s^2$$ and the eigenvalues are $\pm s$.

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Now the next more complicated case when $m=n=2$. The matrix becomes $$\begin{pmatrix}\begin{pmatrix}0&0\\ 0& 0\end{pmatrix} & sC \\ sC^T &f\begin{pmatrix}0 &-1\\ -1 &0\end{pmatrix}\end{pmatrix}.$$ Since $m=n$, we can use the formula $$\det\begin{pmatrix}A & B\\ C& D\end{pmatrix}=\det(AD-BC)$$ and we obtain $$p(\lambda)=\det\left(f\begin{pmatrix}-\lambda & 0\\ 0 & -\lambda\end{pmatrix}\begin{pmatrix}-\lambda & -1 \\ -1 &-\lambda\end{pmatrix}-s^2C^TC\right)\\ =\det\left(f\begin{pmatrix}\lambda ^2 & \lambda\\ \lambda & \lambda ^2\end{pmatrix}-s^2C^TC\right).$$ What can we have for $C^TC$? Let us write $C=(c_{i,j})$ with $c_{i,j}=\pm 1$. Then, $$p(\lambda)=\det\begin{pmatrix}f\lambda^2-s^2(c_{1,1}^2+c_{1,2}c_{2,1})& f\lambda-s^2(c_{1,1}c_{1,2}+c_{1,2}c_{2,2})\\ f\lambda-s^2(c_{2,1}c_{1,1}+c_{2,2}c_{1,2})& f\lambda^2-s^2(c_{2,1}c_{1,2}+c_{2,2}^2)\end{pmatrix}\\ =(f\lambda^2-s^2(c_{1,1}^2+c_{1,2}c_{2,1}))(f\lambda^2-s^2(c_{2,1}c_{1,2}+c_{2,2}^2))-(f\lambda-s^2(c_{1,1}c_{1,2}+c_{1,2}c_{2,2}))(f\lambda-s^2(c_{2,1}c_{1,1}+c_{2,2}c_{1,2}))\\=f^2\lambda^4-f\lambda^2s^2(c_{1,1}^2+2c_{1,2}c_{2,1}+c_{2,2}^2)+s^4(c_{1,1}^2c_{2,1}c_{2,2}+c_{1,1}^2c_{2,2}^2+c_{1,2}^2c_{2,1}^2+c_{1,2}c_{2,1}c_{2,2}^2)-(f^2\lambda^2-f\lambda s^2(c_{1,1}c_{1,2}+c_{1,2}c_{2,2}+c_{2,1}c_{1,1}+c_{2,2}c_{1,2})+s^4(c_{1,1}^2c_{1,2}c_{2,1}+c_{1,2}^2c_{1,1}c_{2,2}+c_{1,2}c_{2,1}c_{1,1}c_{2,2}+c_{1,2}^2c_{2,2}^2)).$$ Each $c_{i,j}^2=1$ so that we obtain $$p(\lambda)=f^2\lambda^4-2f\lambda^2s^2(1+c_{1,2}c_{2,1})+s^4(2+c_{2,1}c_{2,2}+c_{1,2}c_{2,1})-f^2\lambda^2+f\lambda s^2(c_{1,1}c_{1,2}+2c_{1,2}c_{2,2}+c_{2,1}c_{1,1})-s^4(1+c_{1,2}c_{2,1}+c_{1,1}c_{2,2}+c_{1,2}c_{2,1}c_{1,1}c_{2,2})\\ =f^2\lambda^4-f^2\lambda^2-2f\lambda^2s^2a+f\lambda s^2b+s^4c,$$ where $$a=1+c_{1,2}c_{2,1},\\ b=c_{1,1}c_{1,2}+2c_{1,2}c_{2,2}+c_{2,1}c_{1,1},\\ c=2+c_{2,1}c_{2,2}+c_{1,2}c_{2,1}-(1+c_{1,2}c_{2,1}+c_{1,1}c_{2,2}+c_{1,2}c_{2,1}c_{1,1}c_{2,2})=1+c_{2,1}c_{2,2}-c_{1,1}c_{2,2}-c_{1,2}c_{2,1}c_{1,1}c_{2,2}.$$ At this point, a case-by-case study on the possible values for the $c_{i,j}$'s is needed. Let me work out the cases when all of them are one.

Then, $a=2, b=4, c=0$ and $$p(\lambda)=f^2\lambda^4-f^2\lambda^2-4s^2f^2\lambda^2+4fs^2\lambda.$$ One eigenvalue is zero and after dividing out by the corresponding linear factor, we want to find the roots of $$q(\lambda)=f^2\lambda^3-(f^2-4s^2f^2)\lambda+4fs^2.$$ None of the roots is immediate so it is hard to go on from here.

In the case when all entries of $C$ are minus one, $a=2, b=4, c=0$, just as we had before.

If $c_{1,1}=c_{2,2}=1$, $c_{1,2}=c_{2,1}=-1$, $a=2, b=-4, c=0$ which again looks just like before.

You can analyse this situation further but my impression is that we can't say more than that $\lambda=0$ might always be an eigenvalue. If you can, write a little program that tests for you all possible cases for $C$.