Derivation of t2g eg splitting in octahedral field

I am trying to make the derivation of the *d* orbitals splitting under an octahedral field but I am kind of at a loss and don't know of any literature that can provide a more introductory level of this subject (if you know of any, please let me know). 


I am trying to follow the book "Magnetic Oxides" by Gerald F. Dionne but I understand half of what it says in regards to this. 


I know that we want to use the crystal field as a perturbation to the hamiltonian of the center ion, and for that you need to diagonalize the matrix (picture from [here](http://en.citizendium.org/wiki/Crystal_field_theory))
[![enter image description here](https://i.stack.imgur.com/FZHRu.png)](https://i.stack.imgur.com/FZHRu.png)


And that's it, that's where my understanding ends. The subscripts of this matrix is the quantum number *m* and the regular *d* suborbitals are almost all a linear combination of spherical harmonics with different *m*. So how do I diagonalize it? 


The book doesn't do an explicit matrix handling and it basically just tells you from the start what the *eg* and *t2g* orbital are comprised of, I am assuming the notation means something I am unaware of, and that they have different perturbation energies. No explicit equation that makes you understand that is shown. 


I am sorry if it all seems confusing, I am confused. I know the how to derive the octahedral field (in cartesians and spherical harmonics) and how each *d* suborbital is written in spherical harmonics. But I don't know how to obtain the different energy splitting due to the perturbation of the octahedral field, any guidance/book or help is appreciated. Thanks!


PD: I am not a chemist, that's why I don't know of any books where I could read about this subject and why I am probably not following that well the book I was given.




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Edit:


Because of all the recommendations, I have been able to understand more clearly the procedure while looking at it from a matrix perspective rather than a group theory one (completely at a loss there). 


My question is that I am not getting what I should be when obtaining the element of the matrix
[![enter image description here](https://i.stack.imgur.com/SGxSB.png)](https://i.stack.imgur.com/SGxSB.png)


When trying to obtain those numbers in Mathematica I obtain values that don't make sense. I defined the *d* wave function like
[![enter image description here](https://i.stack.imgur.com/JJAc8.png)](https://i.stack.imgur.com/JJAc8.png)


Here you can see one matrix entry as an example
[![enter image description here](https://i.stack.imgur.com/KskVd.png)](https://i.stack.imgur.com/KskVd.png)


When doing the same as above for the four that appear in the image, I get that the factors before the *Dq* need to be 
[![enter image description here](https://i.stack.imgur.com/sW7rD.png)](https://i.stack.imgur.com/sW7rD.png)


Which is not the same as the book.


