# 3 by 3 matrix with an orthogonality constraint

1. Oct 24, 2005

### touqra

This is a paragraph from a book, which I don't understand:
"How many independent parameters are there in a 3x3 matrix? A real 3x3 matrix has 9 entries but if we have the orthogonality constraint,
$$RR^T = 1$$
which corresponds to 6 independent equations because the product
$$RR^T$$ being the same as $$R^TR$$, is a symmetrical matrix with 6 independent entries.
As a result, there are 3 (9-6) independent numbers in R."
I can understand why a real 3x3 matrix has 9 entries. But the sentences after that...I don't understand.

2. Oct 25, 2005

### HallsofIvy

Staff Emeritus
The orginal statement is a bit peculiar. "How many independent parameters are there in a 3x3 matrix?" is answered by the first part of the next sentence. Since a real 3x3 matrix has 9 entries- 9 independent parameters. However, it then talks about the "orthogonality constraint" as if it were talking about orthogonal matrices all along.
Imagine writing out a 3x3 matrix and its transpose, then multiplying them. Since RRT= 1, that gives 9 equations. However, RTR must give the same thing so that not all of those equations are independent. There are 3 equations that say the quantities on the main diagonal are equal to 1 and 3 equations that say the quantities above the main diagonal are equal to 0. The 3 equations that say the quantities below the main diagonal are 0 do not give us anything new because of the symmetry. There are (no more than) 6 independent equations. You could choose 3 of the entries independently (as the parameters) and then solve the 6 equations for the remaining 6 numbers.