On the page P196, "positive definite" should be "positive semi-definite" for covariance as the inner product

[zhfkt]

Describe the mistake P196, It says. "we obtain an inner product. We see that the covariance is symmetric, positive definite,"

According to the wiki on the https://en.wikipedia.org/wiki/Covariance#Relationship_to_inner_products , when the covariance is used as inner product of random variable, it should be positive semi-definite and not "positive definite" . The cov(X,X) = 0 implies that X is a constant random variable other than 0.

The video related with this book on the https://www.coursera.org/lecture/pca-machine-learning/inner-products-of-functions-and-random-variables-optional-luMoJ (04:01) also contains this issue.

Location Please provide the

1. version (bottom of page) Draft (2019-07-27) of “Mathematics for Machine Learning”. Feedback to https://mml-book.com

2. Chapter Chapter 6.

3. page P196

4. line number/equation number The fifth line

Proposed solution Change " positive definite" to positive semi-definite on the pdf (P196) and coursera video (04:01) .

Additional context After searching on the google, I found it seems that the inner product of positive semi-definite is also called "semi norm" or "degenerate inner product" . The author can expand more description on this topic if it is possible. It is also sad to see that the video also contains this issue.

[chengsoonong]

We're unfortunately unable to make major changes to the book. We will add a remark in Chapter 6, after defining the covariance that we assume that covariances are positive definite.

Remark: In this book, we generally assume that covariance matrices are positive definite to enable better intuition. We therefore do not discuss corner cases that result positive semidefinite (low-rank) covariance matrices.

[zhfkt]

Thank you.

Will you add the same remark in the coursera video ?

[mpd37]

We cannot edit the videos directly at this point.