Yes I know that two posts back I said that this would be the last time we would do Pythagoras stuff like this. On the other hand I was very unsatisfied with that post (title: That weird root formula). Also I had wanted to post the math below before but I did not have the time.
All in all since I was horribly bad in the post upon that weird root formula about how to make extra columns, may this short post compensates a bit for that.
This post starts with a column of four real numbers, say a, b, c and d. The goal is to keep on adding extra columns such that all columns are perpendicular to each other. Don’t confuse that with an orthogonal matrix, orthogonal matrices also have all their columns perpendicular but the columns are also all of norm one.
For myself I name matrices that have all their columns perpendicular to each other ‘perpendicular matrices’ but this is not a common thing in math communications as far as I know.
I show you two examples here: First I make an extra column based on the a and b entry of our vector. In the second expansion of the same vector I use the middle two entries b and c.
This should serve as examples that make it as transparant as possible how you must use the +/- chessboard pattern that comes with calculations like this.
For understanding this post it comes in very handy if you have done & understand the general way of crafting the inverse of a square matrix. I think most people will see the brilliant +/- chessboard scheme there for the first time in their lives.

I don’t know much about the history of math, but I like it that the +/- chessboard scheme has no human name attached to it like in “Hilbert space”. I guess this chessboard pattern emerged slowly over the cause of a few decades with contributions of many people. So in the end there was nobody to name it too because this big success just had to many fathers.

Another explanation for the lack of a human name to the famous +/- chessboard pattern is that the person who for the first time chrystal clear wrote out the stuff, this person was not an overpaid professional math professors. But say an amateur just like me. Well in those good old times just like now, the overpaid math professors can’t give credit to such an undesireable person of course…
Yet not all is negative when it comes to professional math professors: They are still very good at telling anybody who wants to hear it that: “We tried but we could not find the three dimensional complex numbers”.

After all that human blah blah blah, why not take a look at the three pictures?

That was it for this post.