In a math book, is it OK to have a diagram wider than text width (so overriding the margins)?
Also, how do I center a wide figure so that it occupies both the left margin and the right margin (not only the right margin) using LaTeX software?
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It depends on the book design. Generally speaking, the width of a text column is kept within certain bounds in order to make the text scannable. A column of text will become much more difficult to read if the text gets wider than the reader can comfortably scan. But many books choose a wider page width so that they can accommodate wider elements such as diagrams, illustrations, and equations. Art books often have text that is less than half the width of the page (or in two columns) so that there is room to show the art at a reasonable size.
In other words, the printable limits of the page are often wider than the width of the text column precisely to accommodate cases like yours. In this case, the equation is not overriding the margin of the page. Rather, it is the text column that is deliberately less wide than the margins of the page.
You can extend visual media as far as you want on the printed page; so long as there is an adequate benefit to the reader (bigger will be more clear, diagram is complex, want subscripts and other symbols more readable...) The only limit is (of course) the page itself and the left-hand gutter.
Recommend asking the technical q re: LaTex separately or diff forum. I just wanted to address that since I'm ostensibly "a-ing the q" -thx.
This question is unclear to me, particularly it is unclear if the diagram is an image, a
tabular or similar environment, a
tikzpicture or anything else.
It is very true what other answers and comments say, this is that it (also) depends on the design of the book and, very much not related, on the technology used for printing. In the next observations I will assume that - e.g. - binding margins are not exceeded.
If you are including your diagram using the command
\includegraphics from the package
graphicx, then you can manually add a negative left margin or, very much more advised, scale (or crop) your image using the options of the command, typically
(see page 46 here).
In a totally different approach, lower-level methods are available in LaTeX to generate a vector graphic or to scale an existing one keeping the captions fonts of the right size.
Good luck with your page layout, post updates if you like!