# How to make a matrix positive definite?

I'm trying to implement an EM algorithm for the following factor analysis model;

$$W_j = \mu+B a_j+e_j \quad\text{for}\quad j=1,\ldots,n$$

where $W_j$ is p-dimensional random vector, $a_j$ is a q-dimensional vector of latent variables and $B$ is a pxq matrix of parameters.

As a result of other assumptions used for the model, I know that $W_j\sim N(\mu, BB'+D)$ where $D$ is the variance covariance matrix of error terms $e_j$, $D$ = diag($\sigma_1^2$,$\sigma_2^2$,...,$\sigma_p^2$).

For the EM algorithm to work, I'm doing dome iterations involving estimation of $B$ and $D$ matrices and during these iterations I'm computing the inverse of $BB'+D$ at each iteration using new estimates of $B$ and $D$. Unfortunately during the course of iterations, $BB'+D$ loses its positive definiteness (but it shouldn't because it is a variance-covariance matrix) and this situation ruins the convergence of the algorithm. My questions are:

1. Does this situation show that there is something wrong with my algorithm since the likelihood should increase at every step of EM?

2. What are the practical ways to make a matrix positive definite?

Edit: I'm computing the inverse by using a matrix inversion lemma which states that:

$$(BB'+D)^{-1}=D^{-1}-D^{-1}B (I_q+B'D^{-1}B)^{-1} B'D^{-1}$$

where the right side involves only the inverses of $q\times q$ matrices.

• It might help to understand better how $BB'+D$ "loses" its positive definiteness. This implies that either $BB'$ or $D$ (or both) are becoming non-positive definite. That's hard to do when $BB'$ is computed directly from $B$ and even harder when $D$ is computed as a diagonal matrix with squares on its diagonal!
– whuber
Mar 15, 2011 at 19:41
• @whuber Typically in FA $q<p$, so $BB'$ isn't ever positive definite. But (theoretically) $BB' + D$ ought to be, assuming that the $\sigma^2_j$'s are all greater than zero.
– JMS
Mar 15, 2011 at 22:03
• This is related to this question: stats.stackexchange.com/questions/6364/… Mar 15, 2011 at 22:12
• @JMS Thank you. I think my comment is still pertinent: $BB'$ can be indefinite, but should still not have any negative eigenvalues. Problems will arise when the smallest of the $\sigma_i^2$ is comparable to numerical error in the inversion algorithm, though. If this is the case, one solution is to apply SVD to $BB'$ and zero out the really small (or negative) eigenvalues, then recompute $BB'$ and add $D$.
– whuber
Mar 15, 2011 at 22:42
• It's got to be small elements in $D$; $I_q + B'D^{-1}B$ should be well-conditioned otherwise since $q<p$
– JMS
Mar 17, 2011 at 18:19

OK, since you're doing FA I'm assuming that $B$ is of full column rank $q$ and $q<p$. We need a few more details though. This may be a numerical problem; it may also be a problem with your data.

How are you computing the inverse? Do you need the inverse explicitly, or can re-express the calculation as the solution to a linear system? (ie to get $A^{-1}b$ solve $Ax=b$ for x, which is typically faster and more stable)

What is happening to $D$? Are the estimates really small/0/negative? In some sense it is the critical link, because $BB'$ is of course rank deficient and defines a singular covariance matrix before adding $D$, so you can't invert it. Adding the positive diagonal matrix $D$ technically makes it full rank but $BB'+D$ could still be horribly ill conditioned if $D$ is small.

Oftentimes the estimate for the idiosyncratic variances (your $\sigma^2_i$, the diagonal elements of $D$) is near zero or even negative; these are called Heywood cases. See eg http://www.technion.ac.il/docs/sas/stat/chap26/sect21.htm (any FA text should discuss this as well, it's a very old and well-known problem). This can result from model misspecification, outliers, bad luck, solar flares... the MLE is particularly prone to this problem, so if your EM algorithm is designed to get the MLE look out.

If your EM algorithm is approaching a mode with such estimates it's possible for $BB'+D$ to lose its positive definiteness, I think. There are various solutions; personally I'd prefer a Bayesian approach but even then you need to be careful with your priors (improper priors or even proper priors with too much mass near 0 can have the same problem for basically the same reason)

• Let me second that in the main part of algorithms, you never want to actually invert a matrix. You may need to at the very end to get the standard estimates though. See this blog post johndcook.com/blog/2010/01/19/dont-invert-that-matrix Mar 16, 2011 at 14:42
• The values of D matrix are getting smaller smaller as the number of iterations increases. Maybe this is the problem as you pointed out. Mar 17, 2011 at 17:41
• @Andy Amos: I'd bet money on it. Like @whuber points out it's almost impossible that $BB'$ would have negative eigenvalues if you're computing it directly, & the zeros (from being rank deficient) should be taken care of by adding $D$ since its positive diagonal - unless some of those elements are really small. Try generating some data from a model where $\sigma_i^2$ are pretty large and $\sum_q B_{iq}^2 \approx \sigma_i^2$. The more data the better so that the estimates should be accurate and stable. That will at least tell you if there's a problem in your implementation.
– JMS
Mar 17, 2011 at 18:17