# Orthogonal Matrices

1. Jan 16, 2009

### war485

1. The problem statement, all variables and given/known data

1. If I got a square orthogonal matrix, then if I make up a new matrix from that by rearranging its rows, then will it also be orthogonal?

2. True/false: a square matrix is orthogonal if and only if its determinant is equal to + or - 1

2. Relevant equations

no equations

3. The attempt at a solution

1. I think it should also be orthogonal since it forms a basis, and the basis would be the same, but just a linear combination of the previous, right?

2. false, its determinant doesn't necessarily ensure it is orthogonal. So, how would/should I correct that statement?

2. Jan 16, 2009

### NoMoreExams

3. Jan 16, 2009

### Unco

An nxn matrix is orthogonal iff its rows form an orthormal basis for $$\mathbb{R}^n$$ (note the symmetry of $$AA^T=A^TA=I$$ for an orthogonal matrix A). The linear independence of a collection of vectors doesn't depend on the order in which you write them, so the rows of the new matrix still form an orthonormal basis.

Just be careful your language: a linear combination of a basis reads as a linear combination of its vectors, which gives just one vector.

4. Jan 16, 2009

### war485

I forgot about the linear independence part of it for #1.

As for #2, I took a counter-example from wikipedia XD
[ 2 0 ]
[ 0 0.5 ]
where its determinant = 1
but the length of each column is not 1 (not orthonormal)
I guess counter-examples should be enough?

Thanks for the help you two. :)

5. Jan 16, 2009

### Unco

The statement #2 is (colloquially) of the form "(property X implies property Y) AND (property Y implies property X)" (*). If all you want to do is show that (*) is false (e.g., if you were asked to prove or disprove the statement), then it suffices to show that property Y does not imply property X.

To show that property Y does not imply property X, it suffices to give an example for which property Y holds but X does not. Why? Because it definitively answers the question as to whether Y implies X. There is no guessing about it!

6. Jan 16, 2009

### war485

Yea, you're right Unco. I need to work on my logic a bit more. I'm very grateful for your help :D

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