Log determinant of positive definite matrices in Matlab

In Bayesian data analysis, the log determinant of symmetric positive definite matrices often pops up as a normalizing constant in MAP estimates with multivariate Gaussians (ie, chapter 27 of Mackay). Oftentimes, the determinant of A will evaluate as infinite in Matlab although the log det is finite, so one can’t use log(det(A)). However, we know that:

$\mathbf{A} = \mathbf{L}\mathbf{L}'$ (Cholesky decomposition)
$|\mathbf{L}| = \prod_i L_{ii}$ (determinant of a lower triangular matrix)
$\log \prod_i x_i = \sum_i \log x_i$ (log rule)

Thus to calculate the log determinant of a symmetric positive definite matrix:

L = chol(A);
logdetA = 2*sum(log(diag(L)));

June 11, 2008

6 responses to “Log determinant of positive definite matrices in Matlab”

james says:

January 30, 2017 at 6:53 pm

Thank you for this! Was having problems with log(det(K)) for optimizing GP hyperparameters.

Reply
Victor Hugo says:

September 18, 2016 at 8:53 am

Thanks for sharing! Does it work for positive semi-definite matrices too?

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- dfgdfg says:
  
  February 8, 2017 at 10:06 am
  
  no
  
  Reply
Astrophysicist at work | The trials and tribulations in the working life of a gravitational wave researcher | PressForward says:

November 2, 2015 at 2:49 pm

[…] of the determinant. The solution, if you’re just wanting the log likelihood anyway, is given by this post, but I repeat the main Matlab commands (Note: the covariance matrix must be positive definite for […]

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The log of a determinant in Matlab | Astrophysicist at work says:

March 6, 2013 at 10:28 am

[…] the determinant. The solution, if you’re just wanting the log likelihood anyway, is given by this post, but I repeat the main Matlab commands (Note: the covariance matrix most be positive definite for […]

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Approximate log determinant of huge matrices « xcorr: computational neuroscience says:

October 6, 2011 at 12:37 am

[…] inference in models with Gaussian priors or likelihoods. They can be quite troublesome to compute; done through the Cholesky decomposition, it’s an operation. It’s pretty much infeasible to compute the exact log det for […]

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