Talk:Linear algebra/Orthogonal matrix

Latex looks broken

@Guy vandegrift: The latex on this page is not actually rendered. It just looks like the latex source when I view the page. Do you know why this might be? AP295 (discuss • contribs) 18:51, 2 January 2024 (UTC)

{{ping|AP295} Thanks! It looks OK to me. I have noticed that I need to purge or refresh a page when I add a new formula recently. I guess that some sort of cache system is used and that the move disrupted it--Guy vandegrift (discuss • contribs) 18:57, 2 January 2024 (UTC)

Sorry, I guess it was a setting I had selected. I was wondering why so much latex did not render, but the setting wasn't really clear about this. I thought it was an input setting rather than a display setting. It looks fine now. AP295 (discuss • contribs) 19:11, 2 January 2024 (UTC)

Comment

The first bullet point (that an orthogonal matrix is a square matrix whose columns form an orthogonal basis) would probably serve better as a definition. As the article suggests, there are many equivalent properties but this is probably the clearest and most informative. I haven't had the opportunity to refresh my knowledge lately, but I had wanted to make a course on linear algebra myself at some point. It's a rewarding topic and I feel it's often made more complicated than it really needs to be. I don't remember being impressed with the relevant articles on Wikipedia, last I checked. On the other hand, I appreciate, e.g. Axler's approach in Linear Algebra Done Right in which takes care not to overcomplicate the essence of the topic. AP295 (discuss • contribs) 19:25, 8 January 2024 (UTC)

Also, orthogonal and orthonormal aren't quite the same thing, though the article seems to suggest that they can be used interchangeably. It should probably be edited to properly reflect the difference between these two terms. Edit: From a cursory search, it looks like several sources define an "orthogonal matrix" like this. Still, using "orthogonal matrix" to mean a square matrix whose columns are orthonormal strikes me as misleading. If one didn't know better, one would just as easily assume an "orthogonal matrix" is simply a matrix whose columns are pairwise orthogonal. Again, I learned using Axler, and he has slightly different terminology, with linear maps and their properties being the point of emphasis rather than matrices per se. I think his approach is a very sensible one. Consider the third bullet point in this resource vis-a-vis Axler's statement of the complex spectral theorem. There's nothing exactly wrong with the third bullet point here and other sources presumably make similar statements, but Axler's statement is strikingly more precise and meaningful, at least in my opinion. I fixed the link to the Wikipedia article, which is frankly quite a terrible article. Wikipedia's math articles usually are, often emphasizing trivialities, jargon and other unnecessary odds/ends (more odds than ends), frequently comprising a rough and disordered tabulation of info without much thought given to structure or communicating the meaning of their contents in general. I'd fix them if I were allowed to. AP295 (discuss • contribs) 08:18, 9 January 2024 (UTC)

Minor repairs

@AP295: I also thought the Wikipedia article was terrible. In fact, I don't like a lot of WP articles on mathematical physics. That's why we need to bend the rules that "Wikiversity is not an encyclopedia". In some fields, encyclopedic articles need to be focused on the reader's current knowledge. Wikipedia tends to offer two articles on something like General Relativity: The one for non-experts contains zero-math, and the one for experts involves more math than I know. The truth is that readers need a separate article every time they complete a course in calculus, linear algebra, or analysis.

I tried to fix the confusion about orthogonal V. orthonormal the best I can. I suspect that there is some disagreement about the definitions, and there is certainly disagreement on whether "orthonormal" is a superior label than "orthogonal". My take on this is that "orthonormal" is the more descriptive term, while "orthogonal" was what they used 40 years ago. Wikipedia redirects w:Orthonormal matrix to w:Orthogonal matrix, and I would hate to read the talk pages on that decision.

I also moved some material here, and propose it stay on the talk pager (or be deleted.) I don't know what I was thinking when i wrote it. Literally. I have no idea what I was trying to do or say. {{Robilar Since orthogonal matrices form a group under multiplication, we can construct a non-trivial orthogonal matrix my multiplying two matrices that are easy to understand. Consider, for example, ${\underline {\underline {C}}}={\underline {\underline {A}}}\cdot {\underline {\underline {B}}}:$

Text I propose to delete $\overbrace {\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\0&0&-1\end{bmatrix}} ^{\underline {\underline {A}}}\cdot \overbrace {\begin{bmatrix}1&0&0\\0&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\0&-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}} ^{\underline {\underline {B}}}=\overbrace {\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{2}}&{\frac {1}{2}}\\-{\frac {1}{\sqrt {2}}}&{\frac {1}{2}}&{\frac {1}{2}}\\0&{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}\\\end{bmatrix}} ^{\underline {\underline {C}}}$

Yours truly 14:32, 9 January 2024 (UTC) Guy vandegrift (discuss • contribs) 14:32, 9 January 2024 (UTC)

Out of curiosity, is there a reason you've created just this page? I notice the containing resource is something of a free-for-all, apparently I made a few edits to it myself a few years ago, shortly after I started editing here. I assume it has to do with your work in physics but I only studied up to electromagnetism, though if I could do it again I'd consider a physics major. AP295 (discuss • contribs) 14:45, 9 January 2024 (UTC)

@Guy vandegrift: Should the equation $Q^{\mathrm {T} }Q^{-1}=Q^{-1}Q^{\mathrm {T} }=I$ in the box on the right not instead be $Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I$? I've also never seen the notation in the second row, why is each matrix doubly underlined? AP295 (discuss • contribs) 17:28, 9 January 2024 (UTC)

On the error: You are correct. Sometimes two rights do make a wrong. I either need the T subscript or the −1, but not both. Regarding the double underline, one professor did that in a two semester course. Underlines for vectors are quicker than arrows, take less space, and allow the generalization to a double underline for a rank-2 tensor (i.e. matrix). Also, this prof also never bothered to use the transpose symbol to distinguish row and column vectors. It was obvious the row vector on the right is the transpose of the column vector on the right in something like ${\underline {a}}\cdot {\underline {\underline {M}}}\cdot {\underline {a}}$ on the left is the transpose of the vector to the right. The fact that only one prof did this motivated the table showing various notations.Guy vandegrift (discuss • contribs) 17:41, 9 January 2024 (UTC)

I suppose you'd keep the transpose in this instance. About your earlier comment, "I also thought the Wikipedia article was terrible. In fact, I don't like a lot of WP articles on mathematical physics. That's why we need to bend the rules that "Wikiversity is not an encyclopedia"." I think wikiversity and wikipedia serve distinct purposes. These concepts could be explained in summary style on Wikipedia, and more fully (or in the style of a course on the subject) here. In my experience, Wikipedia's problem lies with the clique of inexplicably stubborn editors who patrol its math articles. Some of my very first edits were to linear algebra articles on Wikipedia and I ran up against a lot of obstruction even for common-sense stuff. For example the discussions I started in https://w.wiki/8nsK, https://w.wiki/8nsM (the latter was cited to demonstrate an alleged (read: made-up) inability to work with others, in order that I should be blocked indefinitely from Wikipedia, which is so absurd I can hardly wrap my head around it. ) Who's really going to put up with that when they're just trying to improve an article? This is probably why Wikipedia's math and computer science articles remain in a sorry state. At least in the first instance I was able to get my changes through, though it took weeks to persuade the cadre of regulars. In the latter case they refused me outright even though my suggestions were really just common sense if you consider them. There were a few more articles I tried to improve but it's incredibly frustrating when others simply undo your work and refuse to be reasoned with. Funny, it looks like someone agreed with me, a user left a few supporting comments back in 2023, which too were largely ignored and written off by the same people I dealt with. I really can't understand what those editors are thinking, but I digress. It's a common pattern on wikipedia, where you'll see the same perennial comments and suggestions mentioned again and again on a talk page by infrequent editors or one-off IP commenters, serially dismissed by resident editors each time they come up. This is the process that they call "consensus". AP295 (discuss • contribs) 17:53, 9 January 2024 (UTC)