# Thread: Properties of Orthogonal Matrix

1. ## Properties of Orthogonal Matrix

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 $\displaystyle A=A^T$. A is an orthogonal matrix if $\displaystyle AA^T=A^TA=I$ (since $\displaystyle A$ and $\displaystyle A^T$ are symmetric $\displaystyle A$ and $\displaystyle A^T$ commute). And must have an inverse $\displaystyle A^{-1}=A^T$.

Let $\displaystyle R_1,R_2,...R_n$ denote the rows of $\displaystyle A$; then $\displaystyle R^T_1,R^T_2,...R^T_n$ are the columns of $\displaystyle A^T$.

Let $\displaystyle AA^T=C_{ij}$

By matrix multipication: $\displaystyle C_{ij}=R^T_iR^T_j=R_i.R_j$

Therefore $\displaystyle AA^T=I \iff R_i . R_j = \delta_{ij} \iff$ rows of A form an orthonormal set.

My proof is proof is probably not correct. Can anyone show me how to prove this?

2. Originally Posted by demode

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 $\displaystyle A=A^T$. A is an orthogonal matrix if $\displaystyle AA^T=A^TA=I$ (since $\displaystyle A$ and $\displaystyle A^T$ are symmetric $\displaystyle A$ and $\displaystyle A^T$ commute). And must have an inverse $\displaystyle A^{-1}=A^T$.

Let $\displaystyle R_1,R_2,...R_n$ denote the rows of $\displaystyle A$; then $\displaystyle R^T_1,R^T_2,...R^T_n$ are the columns of $\displaystyle A^T$.

Let $\displaystyle AA^T=C_{ij}$

By matrix multipication: $\displaystyle C_{ij}=R^T_iR^T_j=R_i.R_j$

Therefore $\displaystyle AA^T=I \iff R_i . R_j = \delta_{ij} \iff$ rows of A form an orthonormal set.

My proof is proof is probably not correct. Can anyone show me how to prove this?
Your proof is correct but the question is wrong. You have correctly proved that if A is an orthogonal matrix then its row vectors form an orthonormal set. But the question asks you to prove the (false) statement that if A is a symmetric matrix then its row vectors form an orthonormal set.

3. Originally Posted by Opalg
Your proof is correct but the question is wrong. You have correctly proved that if A is an orthogonal matrix then its row vectors form an orthonormal set. But the question asks you to prove the (false) statement that if A is a symmetric matrix then its row vectors form an orthonormal set.
How could the question be wrong?? Because this question is given to us for my course at university...

4. Originally Posted by demode
How could the question be wrong?? Because this question is given to us for my course at university...
I hate to disillusion you, but one of life's hard lessons is that nobody, not even a university professor, is infallible; and nothing that you read, not even printed course material, is necessarily 100% correct.

The matrix $\displaystyle \begin{bmatrix}1&1\\1&1\end{bmatrix}$ is symmetric, but it is not orthogonal and its row vectors do not form an orthonormal set.

5. Thanks, I see! Maybe it's some kind of typo, maybe the word "symmetric" was meant to be "orthogonal" or something like that...

6. Originally Posted by Opalg
I hate to disillusion you, but one of life's hard lessons is that nobody, not even a university professor, is infallible; and nothing that you read, not even printed course material, is necessarily 100% correct.

The matrix $\displaystyle \begin{bmatrix}1&1\\1&1\end{bmatrix}$ is symmetric, but it is not orthogonal and its row vectors do not form an orthonormal set.
Hey, you're not supposed to tell people that! Of course, university professors are infallible!!!

7. Originally Posted by HallsofIvy
Hey, you're not supposed to tell people that! Of course, university professors are infallible!!!

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.

Tonio