Question
Give the upper and lower bounds on the correlation parameter (\rho) for the matrix
$$
C=\begin{pmatrix}
1 & \rho & \cdots & \rho \\
\rho & 1 & \cdots & \rho \\
\vdots & \vdots & \ddots & \vdots
\\
\rho & \rho & \cdots & 1
\end{pmatrix}.
$$
Solution
We write
$$
C = (1-\rho) I + \rho M,
$$
where $M$ is the $n\times n$ matrix of all ones.
Eigenvalues of $M$
The matrix $M$ satisfies:
-
One eigenvalue $n$ with eigenvector $(1,1,\dots,1)^{\top}$. This is because any nonzero eigenvector must satisfy $Mv = \lambda v$, and $\sum_iv_i = \lambda v_i$ for all $1\le i\le n$; $\lambda v_1 = \cdots = \lambda v_n$, therefore $nv_1 = \lambda v_1 \Leftrightarrow \lambda = n$.
-
One eigenvalue $0$ with multiplicity $n-1$.
Thus the eigenvalues of $C = (1-\rho)I + \rho M$ are:
- $1-\rho + \rho n = 1 + (n-1)\rho$, multiplicity $1$,
- $1-\rho$, multiplicity $n-1$.
Positive semidefiniteness constraints
Since $C$ is a correlation matrix, all eigenvalues must be nonnegative:
-
$1-\rho \ge 0 \quad \Longrightarrow\quad \rho \le 1.$
-
$1 + (n-1)\rho \ge 0 \quad \Longrightarrow\quad \rho \ge -\frac{1}{,n-1,}.$