# Ortogonal matrix

1. Jan 16, 2008

### Niles

[SOLVED] Ortogonal matrix

1. The problem statement, all variables and given/known data
I have a 2x2 matrix A:

[alfa beta]
[beta -alfa],

where alfa and beta are real parameters. I have to find out for which values of alfa and beta A is an orthogonal matrix.

3. The attempt at a solution
A matrix is orthogonal if it satisfies Q*Q^T = I.

So I will multiply A with A^T and equal it to I, and I get the condition alfa^2 + beta^2 = 1. Are there any other conditions I need?

2. Jan 16, 2008

### rohanprabhu

ok.. so u have a matrix:

$$\begin{bmatrix} \alpha & \beta \\ \beta & -\alpha \end{bmatrix}$$

on multiplying with it's transpose, you have:

$$\begin{bmatrix} \alpha^2 + \beta^2 & 0\\ 0 & \beta^2 - \alpha^2 \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$

Look at the matrix now, equated with the identity matrix. You've taken one equation correctly. But, does the relation we have now provide you with another equation?

HINT: For two matrices to be equal, all their elements should be equal.

3. Jan 16, 2008

### Niles

Hmm, when multiplying with it's inverse (transpose), I get:

[a^2+b^2 0 ]
[ 0 a^2+b^2].

Last edited: Jan 16, 2008
4. Jan 16, 2008

### HallsofIvy

Yes, you were right the first time. Given only the information that A is a 2x2 orthogonal matrix you only have $\alpha^2+ \beta^2= 1$. That's because there are an infinite number of such matrices, not just one. Any $\alpha$ and $\beta$ satisfying $\alpha^2+ \beta^2= 1$ will give you an orthogonal matrix.

5. Jan 16, 2008

### Niles

Cool, thanks.