Proving a matrix is orthogonal

[member 428835]

Homework Statement

Show that the matrix $P = \big{[} p_{ij} \big{]}$ is orthogonal.

Homework Equations

$P \vec{v} = \vec{v}'$ where each vector is in $\mathbb{R}^3$ and $P$ is a $3 \times 3$ matrix. SO I guess $P$ is a transformation matrix taking $\vec{v}$ to $\vec{v}'$. I also know $\vec{v} = v_i \hat{e}_i$ where $\hat{e}_i$ is the $i$th unit vector.

The Attempt at a Solution

Orthogonal implies $P P^t = I$. $P P^t$ can be wrote in component form as $p_{ij} p_{ji}$. I believe I want to show that $p_{ij} p_{ji} = \delta_{ij}$. After this I'm not really sure how to proceed. Any ideas?

 

[Buzz Bloom]

pijpji=δijp_{ij} p_{ji} = \delta_{ij}.

Hi josh:

The quoted equation is wrong. You want something similar in which you show the index over which you do the sum required in multiplying a row of P by column of P^t.

I'm not really sure how to proceed.

From your problem statement, I am guessing that you were not given a particular matrix you had to show is orthogonal, but rather show a method you can use to show that any given orthogonal matrix is in fact orthogonal. If that is the case, I think your attempted solution (with the correction) is all you need.
Hope this helps,

Regards,
Buzz

 

[member 428835]

Thanks for taking time to reply Buzz! But when you say

Hi josh:
The quoted equation is wrong.

I don't think I wrote what you quoted? I didn't multiply $\delta_{ij}$ by $p$. Perhaps you quoted me while I was editing? But I do agree what I wrote was wrong.

You want something similar in which you show the index over which you do the sum required in multiplying a row of P by column of P^t.

Ok, so to demonstrate $P$ is orthogonal would we have to show $p_{ki}p_{kj} = \delta_{ij}$?

And yea, come to think of it I do think $P$ is a general matrix.

 

[Mark44]

Homework Statement

Show that the matrix $P = \big{[} p_{ij} \big{]}$ is orthogonal.

Homework Equations

$P \vec{v} = \vec{v}'$ where each vector is in $\mathbb{R}^3$ and $P$ is a $3 \times 3$ matrix. SO I guess $P$ is a transformation matrix taking $\vec{v}$ to $\vec{v}'$. I also know $\vec{v} = v_i \hat{e}_i$ where $\hat{e}_i$ is the $i$th unit vector.

I'm confused as to what is the actual problem. Did you put part of the problem statement in the relevant equations? If not, the problem statement, as written, is false.
An arbitrary matrix is not orthogonal.

Also, what does this mean -- $\vec{v} = v_i \hat{e}_i$? In the context of vectors in $\mathbb{R}^3$, it would make more sense to write $\vec{v} = v_1 \hat{e}_1 + v_2 \hat{e}_2 + v_3\hat{e}_3$

The Attempt at a Solution

Orthogonal implies $P P^t = I$. $P P^t$ can be wrote in component form as $p_{ij} p_{ji}$. I believe I want to show that $p_{ij} p_{ji} = \delta_{ij}$. After this I'm not really sure how to proceed. Any ideas?

 

[member 428835]

I'm confused as to what is the actual problem. Did you put part of the problem statement in the relevant equations?

Yes I did, I'm sorry about that!

Also, what does this mean -- $\vec{v} = v_i \hat{e}_i$? In the context of vectors in $\mathbb{R}^3$, it would make more sense to write $\vec{v} = v_1 \hat{e}_1 + v_2 \hat{e}_2 + v_3\hat{e}_3$

I was using Einstein notation, so it means exactly the sum that you wrote in the end.

 

[Mark44]

Also, what does this mean -- $\vec{v} = v_i \hat{e}_i$? In the context of vectors in $\mathbb{R}^3$, it would make more sense to write $\vec{v} = v_1 \hat{e}_1 + v_2 \hat{e}_2 + v_3\hat{e}_3$

I was using Einstein notation, so it means exactly the sum that you wrote in the end.

Wouldn't the right side be shown in brackets, like this?
$[ v_i \hat{e}_i]$
This is similar to the shorthand notation $[p_{ij}]$ that you used in the OP to represent all of the entries of matrix P.

In any case, what is the exact problem statement? From what you've provided so far, I don't see how one can show that an arbitrary matrix is orthogonal.

 

[member 428835]

I'm confused as to what is the actual problem.

This was also given, but I didn't include it because it seemed like it was of no help:

If a vector $\vec{v}$ has coordinates $v_i$ with respect to a basis $\vec{e_i}$ , the transformation rule will tell us the coordinates of the same vector $\vec{v}$ with respect to a different basis $\vec{e_i}'$ . Let $v_i'$ denote the coordinates of $\vec{v}$ with respect to $\vec{e_i}'$. Our goal is to find the transformation rule governing $v_i'$ and $v_i$.

Since $\vec{e_i}$ is a basis, it is possible to find a unique set of 9 numbers, $p_{ij}$ such that $\vec{e_i}' = p_{ij}\vec{e_j}$.

 

[member 428835]

In any case, what is the exact problem statement? From what you've provided so far, I don't see how one can show that an arbitrary matrix is orthogonal.

I have posted the notes that correspond to $P$.

 

[Mark44]

From post #1:

Orthogonal implies $P P^t = I$.

This also implies that $P^{-1} = P^t$.

Maybe I'm missing something, but I don't see anything in the problem description that would lead me to believe that the matrix is orthogonal. You have Pv = v', but you don't show anything about v', other than it is a vector in R³.

 

[member 428835]

From post #1: This also implies that $P^{-1} = P^t$.

Maybe I'm missing something, but I don't see anything in the problem description that would lead me to believe that the matrix is orthogonal. You have Pv = v', but you don't show anything about v', other than it is a vector in R³.

I totally agree. To me it looks as though we are given a square $3 \times 3$ matrix and asked to show this property is true. I'll ask the professor about it, I just wanted to see if anyone else picked up on something I did not. Thanks for your help Mark44!