Orthogonal Matrix

1. Nov 28, 2006

sherlockjones

Assume that $$I$$ is the $$3\times 3$$ identity matrix and $$a$$ is a non-zero column vector with 3 components. Show that:

$$I - \frac{2}{| a |^{2}}aa^{T}$$ is an orthogonal matrix?

My question is how can one take the determinant of $$a$$ if it is not a square matrix? Is there a flaw in this problem?

Thanks

Last edited: Nov 28, 2006
2. Nov 28, 2006

OlderDan

I assume you are referring to the $$| a |^{2}$$ and I also assume that is the inner product (dot product) for the vector. It's just a normalization factor

3. Nov 28, 2006

HallsofIvy

Staff Emeritus
Yes. |a| is not a "determinant", it is the length of the vector a.

4. Nov 28, 2006

Tomsk

Remember that $aa^{T}$ does NOT equal $a^{T}.a$, the scalar product. Use matrix multiplication. You don't need to find the determinant of anything either.