Find the principal components of a $4\times 4$ matrix

Question

I want to find the principal components of the matrix

$$ A=\pmatrix{1,\rho,\rho, \rho\\\rho,1,\rho,\rho\\\rho,\rho,1,\rho\\ \rho,\rho,\rho,1} $$

I assume $\rho$ is correlation. To find the principal components, I believe you have to find the eigenvectors $v_i$ right? Is there a quicker method for this matrix as it seems quite tedious?

Answer 1

You can write $A = (1-\rho)I + \rho vv^T$ where $v = \begin{bmatrix}1 & 1 & 1 & 1\end{bmatrix}^T$.

The matrix $\rho vv^T$ is rank-$1$, so it should be easy to find the eigenvectors and eigenvalues. Adding $(1-\rho)I$ increases the eigenvalues by $1-\rho$ and leaves the eigenvectors unchanged.

Answer 2

$A=\rho J + (1-\rho)I$, so it has the same eigenvectors as $J$, the matrix of all ones.

This means the eigenvector $(1,\ldots,1)$ and the $n-1$ eigenvectors $(1,0,\ldots,0,-1,0,\ldots,0)$.
Plugging into $A$ gives the eigenvalues:
$(1,1,1,1)\mapsto 1+3\rho$
$(1,-1,0,0)\mapsto 1-\rho$
$(1,0,-1,0)\mapsto1-\rho$
$(1,0,0,-1)\mapsto1-\rho$.