# Theorems helpful for largest-eigenvalue lower bound of correlation matrix

Need suggestion of theorems for lower bound of the largest eigenvalue of the correlation matrix (symmetric, diagonal all being one, all other values in range $[-1,1]$).

Of course the eigenvalue has to be non-negative since the matrix is positive semi-definite, but need help with a tighter lower bound.

• If it's an $n\times n$ matrix where every entry has modulus at most $1$, then the largest eigenvalue is at most $n$. This is attained when every entry is $1$. Commented Jan 13, 2018 at 4:48

Correlation matrix $C$ being semi-definite positive, its eigenvalues are : $0 \leq \lambda_1 \leq \cdots \leq \lambda_n.$

The trace of $C$ being $n$, we thus have: $\lambda_n \geq 1$.

This bound $1$ is attained each time $C=I_n$ (identity matrix), for example by taking any pair of independently generated $n$-dimensional vectors.

Thus $1$ is the answer to your question.

• First, thank you for your answer. Second, maybe I misunderstood your answer, but I stated "all other values in $[-1,1]$", i.e. the matrix is not ${\bf 1}^T{\bf 1}$, which is a special case.
– Tony
Commented Jan 13, 2018 at 14:38
• I have understood that all other values values are in $[-1,1]$, but I realize now that I have misunderstood that you wanted a lower bound for $\max \lambda_k$. This lower bound is 1. I modify my answer Commented Jan 13, 2018 at 16:14
• Correction done. Sorry for having misunderstood your question... Commented Jan 13, 2018 at 16:34