Its unlikely you have enough data to directly attack the correlation matrix you describe. If you obtained the MxM (M=6000) correlation matrix from a data matrix of M rows, N columns, I'm guessing N

For me, the problem seems straight forward. You had data (for e.g M rainfall stations or M loan etc.) and you performed correlation operation getting (M*M) matrix.

Now you want to segregate that into the homogeneous cluster.

Although, I am not at all satisfied with the linear correlation because I am not sure whether it would be able to capture the variability of data. Anyhow, I am sure you can take care of this.

For clustering, I would suggest you go for K-means unsupervised algorithm or Self-organizing map (SOM) algorithm.

Both are very simple and easy to apply.

See here:

 https://www.researchgate.net/publication/305222653_Wavelet_Spectrum_and_Self-Organizing_Maps-Based_Approach_for_Hydrologic_Regionalization_-a_Case_Study_in_the_Western_United_States

https://www.researchgate.net/publication/277715753_Hydrologic_Regionalization_using_wavelet_based_multiscale_entropy_method

Article Wavelet Spectrum and Self-Organizing Maps-Based Approach for...

Article Hydrologic Regionalization using wavelet based multiscale en...

Thanks all for their suggestion. AI have taken data of S&P 500 Index wherein I have calculated the return of 500 companies and now trying to cluster these companies based on their return correlation matrix. I have tried a few unsupervised clustering techniques like k-means, hierarchical, affinity, HDBSCAN but no technique is giving me a desired or good result