What do the following eigenvalues/eigenvectors tell me regarding the bands in the following LANDSAT TM image?

I am struggling to understand the interpretation for eigenvalues for a false-color NIR LANDSAT TM image. The image is shown below. (Note: I’m using ENVI Classic, but this is a general question and not necessarily software specific).

To see how the bands correlate to one another, I generated the correlation matrix and the eigenvector matrix, as seen below:

Correlation    Band 1      Band 2      Band 3      Band 4      Band 5      Band 6
     Band 1  1.000000    0.961168    0.939219   -0.199623    0.458798    0.694562
     Band 2  0.961168    1.000000    0.960990   -0.197893    0.509627    0.718807
     Band 3  0.939219    0.960990    1.000000   -0.338474    0.524662    0.781865
     Band 4 -0.199623   -0.197893   -0.338474    1.000000    0.214992   -0.107974
     Band 5  0.458798    0.509627    0.524662    0.214992    1.000000    0.876531
     Band 6  0.694562    0.718807    0.781865   -0.107974    0.876531    1.000000

Eigenvector    Band 1      Band 2      Band 3      Band 4      Band 5      Band 6
     Band 1 -0.286913   -0.180856   -0.327835   -0.045758   -0.754041   -0.454781
     Band 2 -0.184842   -0.107840   -0.256091    0.914194    0.209379   -0.095033
     Band 3 -0.585081   -0.319049   -0.435785   -0.377332    0.467093    0.073647
     Band 4  0.175129    0.203321    0.011737   -0.120604    0.408026   -0.864189
     Band 5  0.693180   -0.204841   -0.679428   -0.068775    0.022622    0.103332
     Band 6 -0.173279    0.877821   -0.418657   -0.022377   -0.049092    0.145671

Bands 1-3 are highly correlated in the correlation matrix because they are in the visible spectrum. There is less correlation in the eigenvector matrix; what exactly is the purpose of looking at the eigenvector matrix, and what is a practical interpretation for this data?