Let $t$ be the number of days (time periods), and let $p$ be the number of assets. You have $t=1000$ and $p=10000$. For any given dataset, it is assumed that the sample covariance matrix $\mathbf{C}$ accurately represents the population covariance matrix $\boldsymbol{\Sigma}$, however, as $p \rightarrow t$ or if $p > t$ (as in your case), the eigenvalues become unreliable and can also take on a value of zero, resulting in lack of positive definiteness. With high-dimensional datasets becoming more popular, there is greater potential for the number of dimensions to approach the sample size ($p \rightarrow t$), leading to biased eigenvalues of $\mathbf{C}$ and $\mathbf{R}$. Certainly, there will be $p-t$ zero eigenvalues whenever $p>t$ and one zero eigenvalue whenever $p=t$.
You can use singular value decomposition (SVD), which will extract the singular values (eigenvalues) along with the remaining singular values. If $\mathbf{X}$ is your return matrix ($t$ rows, $p$ columns) then use the R syntax below to look at the eigenvalues ("eigvals") from eigendecomposition versus the singular values ("s") from SVD:
R=cor(X);
p <- ncol(X);
t <- nrow(X);
lambdae <- eigen(R);
eigvals <- as.vector(lambdae$values);
E<-as.matrix(lambdae$vectors);
s<-svd(R)
s$d
Last, if you are going to do anything with your data, you might perform dimension reduction by using the eigenvectors to represent your data for dimensions that have non-zero eigenvalues, as they are uncorrelated. You could also use PCA after extracting the singular-values, and ignore the zero eigenvalues. The loadings with the principal components will represent correlation between the original 10000 assets and the reduced orthogonal (non-correlated) dimensions.
