Covariance Matrix and Correlation Matrix - Singularity

If a covariance matrix is non-singular, does this implies that correlation matrix is also non-singular.

My guess is it depends on mean vector in $K_{X} = R_{X} - m_X.{m_X}^H$

Not sure though.

• The answer to the question in the first sentence is Yes; covariances and correlation coefficients do not depend on the values of the means. – Dilip Sarwate May 22 '16 at 14:56
• @DilipSarwate. If covariances and correlation coefficients do not depend on the values of the means, does this not imply that singularity of either of them is independent of the other? Therefore giving the answer of No for the given question. i.e I can have a singular Kx but non-singular Rx? – urwaCFC May 22 '16 at 17:06
• $R_X$ is not called the correlation matrix in statistical circles. The standard definition of correlation matrix is $\rho_X = \sigma^{-1} K_X \sigma^{-1}$ where $\sigma^{-1} = \operatorname{diag}(\sigma_1^{-1},\sigma_2^{-1},\cdots,\sigma_n^{-1})$, that is, we get $\rho_X$ from $K_X$ by dividing each $i$-$j$-th entry of $K_X$ by $\sigma_i\sigma_j$. – Dilip Sarwate May 22 '16 at 23:10