So, I tried to prove this question in two parts (A) if A is a symmetric matrix then it is orthogonal (B) since A is orthogonal, its row vectors must form an arthonormal set. But I don't think it is possible to prove part (A).
Anyway here's my proof so far:
If A is symmetric then . A is an orthogonal matrix if (since and are symmetric and commute). And must have an inverse .
Let denote the rows of ; then are the columns of .
By matrix multipication:
Therefore rows of A form an orthonormal set.
My proof is proof is probably not correct. Can anyone show me how to prove this?
The matrix is symmetric, but it is not orthogonal and its row vectors do not form an orthonormal set.
Well, many times that's true, but not always. Being an instructor at my alma mater , some time ago already, I had to convince students that not all that's written is infallible, and at long last I told them directly and unmistakenly at the beginning of some course:
"Listen you all: you must check your notes and books and think and etc....because any mistake that that I, or any teacher, book, or oracle whatsoever may commit when teaching you people stuff is your only and unique responsibility!"
The above, of course, is a little hard to swallow for freshman and even sometimes for sophomore students, but that way they stopped trying to justify their nonsenses by saying : "the teacher said this, or that book said so...".
If the teacher makes a mistake it is the serious, commited student's task and duty to find it and mend it...at least by exams time, if not before.