Linear Algebra - Find Orthogonal Matrix Q that diagonals

YoshiMoshi

Homework Statement

I'm told to find the matrix Q of the matrix A

The Attempt at a Solution

So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3
S = [X_1, X_2, X_3] but when I form this matrix with the values for X_1, X_2, X_3 I'm not able to factor out a 1/3. Does anyone know where this value comes from?
This is the solutions

and this is my attempt, again I would get the same answer but have no idea were the 1/3 came from.

thanks for any help

Homework Helper
An orthogonal matrix is a unitary matrix. Therefore ##Q^*Q = QQ^*=I##. This is why the vectors making up the column of ##Q## must be orthonormal - 1/3 is a normalization constant.

YoshiMoshi
How do I calculate this normalization constant? Thanks for your help. Like I'm exactly sure which vector I'm normalizing and getting 1/3. But I know to normalize a vector it's

V/||V||

Homework Helper
A vector is said to be normalized if its norm equal unity.

YoshiMoshi
So it's just

1/sqrt((1+1+1)^2)?

Homework Helper
1/sqrt((1+1+1)^2)?
I don't understand what you want to say there.
Suppose you have three component vectors ##u = (a,b,c)^T##. In order to make it normalized you have to add a common constant by yourself to the vector: ##u = \gamma (a,b,c)^T##. By requiring its norm to be equal to one, namely ##\gamma^2(|a|^2 + |b|^2 + |c|^2) = 1##. This way you can find ##\gamma## in terms of the components of ##u##. For a general unitary matrix, ##\gamma## can be complex but since here the problem asks for an orthogonal matrix, ##\gamma## must be real.

YoshiMoshi
Oh I see that makes more since to me know. So it's for the whole Matrix S? When I square all the terms in S and sum each row individually I get 9, so the constant must be 1/sqrt(9) = 1/3.

What if each row doesn't sum up to the same value like it did in this case?

Homework Helper
Then you cannot pull the common normalization constant out of the matrix's bracket. Nothing wrong with that.

YoshiMoshi
ah ok thanks for your help