Derivations of Equations in PRML by C. Bishop

As I read through PRML (Feb 2006 edition), I made notes on derivations of non-trivial equations. There are many solution manuals available for the exercises, but none for the equations in the text itself. So I thought my solutions might help others to push through this difficult book. Bishop assigned many equations in the text as exercises, and I did not include them in my document. The following is the list of the solution manuals for exercices available on the internet.

1. PRML Solutions to Exercises: Tutor’s Edition
2. PRML Solutions to Exercises: Web Edition
3. Solution Manual for PRML by Zhengqi Gao

I used the same mathematical notation as in PRML except for t , which is a column vector of a list of observations in this document.

1. t: a binary-categorical target value.
2. t : a vector for multi-categorical target values. ${t_1, t_2, \cdots , t_M }$, where M is the dimension of the feature space.
3. $\overline{\textbf{t}}$: a vector for a list of observations of binary-categorical target values. ${t_1, t_2, \cdots , t_N }$, where N is the number of observations.

Reference:
The Matrix Cookbook (Nov 2012) by K. B. Peterson and M. S. Pederson (https://www2.imm.dtu.dk/pubdb/pubs/3274-full.html)