<p>The "trick" you are referring to is a property of the trace of a product of matrices, namely</p>

<p>$${\rm tr}(ABC) = {\rm tr}(BCA)$$</p>

<p>assuming the dimensions are conformable of course.  </p>

<p>Now, note that the dimension of $(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)$, for each observation, is $1 \times 1$. So trivially,</p>

<p>$${\rm tr}\Big[(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)\Big] = (x_n-\mu)^T\Sigma^{-1} (x_n-\mu) $$</p>

<p>But since also, from the mentioned property of the trace,</p>

<p>$${\rm tr}\Big[(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)\Big] = {\rm tr}\Big[\Sigma^{-1} (x_n-\mu)(x_n-\mu)^T\Big]$$</p>

<p>combining one gets the alternative expression for the mutlivariate normal log-likelihood.</p>

CC BY-SA 3.0
2015-12-07T01:10:33.907Z
2015-12-07T01:10:33.907Z
Alecos Papadopoulos
<p>I am reading econometrics lecture notes of a Japanese professor. To explain MLE estimation he uses multivariate normal distribution. <a href="https://i.sstatic.net/aZFU3.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/aZFU3.jpg" alt="mle"></a> </p>

<p>As you can see, the first line is the loglikelihood function and the second line is proportional to the first line equation. I cannot get my head around this. How and why the first equation can be written as the second one using trace of a matrix. I heard about "trace trick" and this must be the application of this trick, but I need help to understand it and possibly use it.
Thanks</p>

<econometrics><mathematical-economics><statistics>
MLE using multivariate normal distribution
london
# MLE using multivariate normal distribution

(Asked by: username_0 on 2015-12-06)

I am reading econometrics lecture notes of a Japanese professor. To explain MLE estimation he uses multivariate normal distribution. [![mle](https://i.sstatic.net/aZFU3.jpg)](https://i.sstatic.net/aZFU3.jpg)

As you can see, the first line is the loglikelihood function and the second line is proportional to the first line equation. I cannot get my head around this. How and why the first equation can be written as the second one using trace of a matrix. I heard about "trace trick" and this must be the application of this trick, but I need help to understand it and possibly use it.
Thanks

**Tags:** econometrics, mathematical-economics, statistics

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## 1 answers

### Answer by username_1 on 2015-12-07. Score: 4 | Accepted answer

The "trick" you are referring to is a property of the trace of a product of matrices, namely

$${\rm tr}(ABC) = {\rm tr}(BCA)$$

assuming the dimensions are conformable of course.

Now, note that the dimension of $(x\_n-\mu)^T\Sigma^{-1} (x\_n-\mu)$, for each observation, is $1 \times 1$. So trivially,

$${\rm tr}\Big[(x\_n-\mu)^T\Sigma^{-1} (x\_n-\mu)\Big] = (x\_n-\mu)^T\Sigma^{-1} (x\_n-\mu) $$

But since also, from the mentioned property of the trace,

$${\rm tr}\Big[(x\_n-\mu)^T\Sigma^{-1} (x\_n-\mu)\Big] = {\rm tr}\Big[\Sigma^{-1} (x\_n-\mu)(x\_n-\mu)^T\Big]$$

combining one gets the alternative expression for the mutlivariate normal log-likelihood.


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