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Using Cholesky decomposition to compute covariance matrix determinant

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0

$begingroup$

I am reading through this paper to try and code the model myself. The specifics of the paper don't matter, however in the authors matlab code I noticed they use a Cholesky decomposition instead of computing the determinant of a covariance matrix directly.

Specifically, the author has

$$logdet(Sigma) = 2 sum_i log [ diag(L)_i ]$$

where $L$ is the lower triangular matrix produced by a Cholesky decomposition of the covariance $Sigma$.

I tested this out myself using various covariance matrices and found the relation above always works to within 14 decimal places (it's probably just a machine precision issue). I believe the author uses a cholesky decomposition because it is slightly faster to compute than computing the determinant directly (at least when I timed it on my machine).

My question is, why does this relation hold true? I couldn't find any references in the paper / code, or any material online.

determinant matrix-decomposition covariance

share|cite|improve this question

asked Mar 22 at 15:37

PyRsquaredPyRsquared

1444

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What does "computing the determinant directly" mean in this context? If you are using a library, the routine to compute determinants might well be using something like Gaussian elimination or Cholesky decomposition, or whatever, and not summing over all permutations or expanding by minors.
  $endgroup$
  – kimchi lover
  Mar 22 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $Sigma=LL^T$ so you take det, use eigenvalues, then log and it follows immediately.
  $endgroup$
  – Michal Adamaszek
  Mar 22 at 15:52
  

  
  
  
  

add a comment | 

0

$begingroup$

I am reading through this paper to try and code the model myself. The specifics of the paper don't matter, however in the authors matlab code I noticed they use a Cholesky decomposition instead of computing the determinant of a covariance matrix directly.

Specifically, the author has

$$logdet(Sigma) = 2 sum_i log [ diag(L)_i ]$$

where $L$ is the lower triangular matrix produced by a Cholesky decomposition of the covariance $Sigma$.

I tested this out myself using various covariance matrices and found the relation above always works to within 14 decimal places (it's probably just a machine precision issue). I believe the author uses a cholesky decomposition because it is slightly faster to compute than computing the determinant directly (at least when I timed it on my machine).

My question is, why does this relation hold true? I couldn't find any references in the paper / code, or any material online.

determinant matrix-decomposition covariance

share|cite|improve this question

asked Mar 22 at 15:37

PyRsquaredPyRsquared

1444

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What does "computing the determinant directly" mean in this context? If you are using a library, the routine to compute determinants might well be using something like Gaussian elimination or Cholesky decomposition, or whatever, and not summing over all permutations or expanding by minors.
  $endgroup$
  – kimchi lover
  Mar 22 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $Sigma=LL^T$ so you take det, use eigenvalues, then log and it follows immediately.
  $endgroup$
  – Michal Adamaszek
  Mar 22 at 15:52
  

  
  
  
  

add a comment | 

$begingroup$

I am reading through this paper to try and code the model myself. The specifics of the paper don't matter, however in the authors matlab code I noticed they use a Cholesky decomposition instead of computing the determinant of a covariance matrix directly.

Specifically, the author has

$$logdet(Sigma) = 2 sum_i log [ diag(L)_i ]$$

where $L$ is the lower triangular matrix produced by a Cholesky decomposition of the covariance $Sigma$.

I tested this out myself using various covariance matrices and found the relation above always works to within 14 decimal places (it's probably just a machine precision issue). I believe the author uses a cholesky decomposition because it is slightly faster to compute than computing the determinant directly (at least when I timed it on my machine).

My question is, why does this relation hold true? I couldn't find any references in the paper / code, or any material online.

determinant matrix-decomposition covariance

share|cite|improve this question

asked Mar 22 at 15:37

PyRsquaredPyRsquared

1444

$endgroup$

share|cite|improve this question

asked Mar 22 at 15:37

PyRsquaredPyRsquared

1444

share|cite|improve this question

asked Mar 22 at 15:37

PyRsquaredPyRsquared

1444

asked Mar 22 at 15:37

PyRsquaredPyRsquared

1444

asked Mar 22 at 15:37

PyRsquaredPyRsquared

1444