Question

Show that the correlation matrix of any random vector X is nonnegative definite, where the correlation matrix $$R\in \mathbb{R}^{m \times m}$$ is defined by $$R_{i,j}$ =$$\frac{Cov(X_i,X_j)}{\sqrt{Var(X_i)Var(X_j)}}$ , $i,j=1,...,m.$ (Assume we know that the covariance matrix of X denoted $\sum$$_X$ is defined by $\sum$$_X$$=\mathbb{E}[(X-\mathbb{E}X)(X-\mathbb{E}X)^T]$  is nonnegative definite, and $(\sum$$_X)$$_{i,j}=Cov(X_i,X_j)=\mathbb{E}[(X_i-\mathbb{E}X_i)(X_j-\mathbb{E}X_j)]$.

Answer 1