What is 9c asking in this question? (Orthogonal Matrix)

• April 7th 2013, 11:20 PM
lukasaurus
What is 9c asking in this question? (Orthogonal Matrix)
http://i.imgur.com/VymorpE.png
I have done a) yes it is (checking that u1.u2, u2.u3 and u1.u3 = 0 & b) by creating an augmented matrix and reducing it to RREF, giving me a coordinate vector.

But what is C asking? I have a feeling this assignment was used last year, but I know that they modified the curriculum this year, and it's not in any of the material (at least, not this notation). It looks like it is asking me to create a new Matrix, consisting of three vectors divided by their length. But I'm not sure. I've been through all the material and cannot find any examples of this? Any help?
• April 8th 2013, 12:38 AM
Gusbob
Re: What is 9c asking in this question? (Orthogonal Matrix)
Quote:

Originally Posted by lukasaurus
http://i.imgur.com/VymorpE.png
I have done a) yes it is (checking that u1.u2, u2.u3 and u1.u3 = 0 & b) by creating an augmented matrix and reducing it to RREF, giving me a coordinate vector.

But what is C asking? I have a feeling this assignment was used last year, but I know that they modified the curriculum this year, and it's not in any of the material (at least, not this notation). It looks like it is asking me to create a new Matrix, consisting of three vectors divided by their length. But I'm not sure. I've been through all the material and cannot find any examples of this? Any help?

You are right about the interpretation of $M$.

An orthogonal matrix $A$ satisfies the equation $AA^T=A^TA=I$. That is, $A^T=A^{-1}$. Just multiply $M$ with its transpose to see if you get $I$ (you should). Your result in part a) guarantees your off diagonal entries are zero, and the $(i,i)^{th}$ entry of $M^TM$ is given by $\left(\frac{u_i}{\| u_i\|}\right)^T\frac{u_i}{\| u_i\|}\right=\frac{u_i\cdot u_i}{\|u_i\|\cdot \|u_i\|}=\frac{\|u_i\|^2}{\|u_i\|^2}=1$

Alternatively, you may use the other definition of orthogonal matrix: a square matrix whose columns and rows are orthogonal unit vectors. Substitute the exact values in and check that this is true for each row (you've already shown orthogonality for columns, and the vectors are, by definition, unit).
• April 8th 2013, 01:26 AM
lukasaurus
Re: What is 9c asking in this question? (Orthogonal Matrix)
I went ahead with my understanding of M (glad to see it was right), and came up with, for c)
M =
1/sqrt(2), -1/sqrt(18), 6
0, 4/sqrt(18), 3
1/sqrt(2), 1/sqrt(18), -6

by finding the dot product of m1.m2, m2.m3 and m1.m3 and verifying they all = zero, this shows it is orthogonal right?

If you have time, could you check my math here? Also From , my coordinate vector waas -5/2, 3/2, -2
• April 8th 2013, 01:41 AM
Gusbob
Re: What is 9c asking in this question? (Orthogonal Matrix)
You didn't normalise your last column. It should be $\frac{1}{3}(2,1,-2)$. The name 'orthogonal matrix' is a bit of a misnomer. You need to show the columns AND rows are orthogonal (with other columns/rows respectively) AND unit. So the dot product of distinct rows need to be zero as well. Also, the dot product of each row/column with itself needs to be 1.
• April 8th 2013, 02:24 AM
lukasaurus
Re: What is 9c asking in this question? (Orthogonal Matrix)
Got it :) Thanks! I was wondering why MM^T wasn't equal to I, since the question is obvious that it should be.