one of Eigenvalues of covariance matrix is negativ

Question

I have a data set x. And I use cov(x) to calculate the covariance of x. I want to calculate the inverse square root of cov(x). But I get negative eigenvalue of cov(x).

Here is my code

S11=cov(x)
S=eigen(S11,symmetric=TRUE)
R=solve(S$vectors %*% diag(sqrt(S$values)) %*% t(S$vectors))

This is the eigenvalue of S.

c(0.897249923338732, 0.814314811717616, 0.437109871173458, 0.334921280373883, 
0.291910583884559, 0.257388456770167, 0.166787180227719, 0.148268784967556, 
0.121401731579852, 0.0588333377333529, 0.0519459283467876, 0.0472867806813002, 
0.0438199555429584, 0.0355421239839632, 0.0325106968911777, 0.0282860419784165, 
0.0222240269478354, 0.0174657163114068, 0.012318267910606, 0.00980611646284724, 
0.00969450391092417, 0.00804912897151307, 0.00788628666010145, 
0.00681419055130702, 0.00664707528670254, 0.00591471779140177, 
0.00581608875646686, 0.0057489828718098, 0.00564645095578336, 
0.00521029715741059, 0.00503304953884416, 0.0048677189522647, 
0.00395692706081966, 0.00391665618240403, 0.00389825739725093, 
0.00383611535401152, 0.00374242176786387, 0.0035160324422885, 
0.00299245160843966, 0.0029501156885799, 0.00289484923017341, 
0.00287327878694529, 0.0028447265712214, 0.00274130080219099, 
0.00273159993035393, 0.00265595612239575, 0.00261856622830277, 
0.0020004125628823, 0.00199834766485368, 0.00199579695856402, 
0.00198945452395265, 0.00197999810684363, 0.00195954105720554, 
0.00195502875017394, 0.00194143254092788, 0.00192530399875842, 
0.00191287435824908, 0.00187418676921454, 0.00184304720875652, 
0.00181132707713659, 0.00167004122321738, 0.00132136106130093, 
0.001001001001001, 0.001001001001001, 0.001001001001001, 0.00100089827907564, 
0.000999613336959707, 0.000999285885989665, 0.000995390174780253, 
0.000990809217795241, 0.000987333916025995, 0.000984260717691378, 
0.000982735942052615, 0.000971684328336702, 0.000964125499180901, 
0.000961900381008093, 0.000947883827257506, 0.000922293473088298, 
0.000862086463606162, 0.000829687294735196, 0.000732694198613695, 
1.95782839335209e-17, 4.13905030077713e-18, 2.02289095736911e-18, 
8.72989281345777e-19, 3.79161425300691e-19, -7.97468731082902e-20)

Answer 1

While in theory an estimated covariance matrix must be positive (semi-)definite, i.e. no negative values, in practice floating-point error can violate this. To me it's no surprise that an 87-by-87 matrix could have a tiny negative (about -1*10^(-19)) eigenvalue.

Depending on what you want to do, you could use ?nearPD from the Matrix package to force your covariance matrix to be positive-definite:

Compute the nearest positive definite matrix to an approximate one, typically a correlation or variance-covariance matrix.

Also, it will probably be more efficient to compute the Cholesky decomposition (?chol) of your matrix first and then invert it (this is easy in principle -- I think you can use backsolve()).