# “Ambiguous up to scale” , Explanation required.

I am reading "Computer Vision: Models, Learning, and Inference" in which author writes at several points (like on page 428-429) that although matrix A seems to have 'n' degree of freedom but since it is ambiguous up to scale so it only has 'n-1' degree of freedom? Can anyone explain what this thing means? Why one degree of freedom is decreased? Although it is in computer vision book but I think the answer is related to some properties of Linear algebra.

• Is it true in this case that matrices $A$ and $\lambda A$ represent the same thing (for $\lambda$ a number)? – Blazej Sep 28 '15 at 9:49
• @blazej yes, both represent same? what does this imply? how a degree of freedom is lost? – Ameer Hamza Sep 28 '15 at 10:13

It might be illuminating to learn about projective spaces (just the idea.) I will explain. Define $\mathbb{RP}^1$ (so called projective line) as set of all straight lines in a plane going through origin. Any such line is uniquely determined by specifying one of its points $(x,y) \neq 0$. However, for any nonzero $\lambda$ points $(x,y)$ and $(\lambda x, \lambda y)$ are equivalent in the sense that they determine the same line. Clearly $\mathbb {RP}^1$ is one dimensional. You can parameterize all of lines (except horizontal one i.e. the one with $y=0$) by just giving x coordinate of a point such that $y=1$. Another way is to use angle of inclination of your line to $x$ axis as a coordinate. Your example is similar. Since scaling of matrix doesn't change anything you are really interested in set of all directions in space of matrices. In principle you could specify this direction by giving $n-1$ "angles "of some sort; or if your matrices are nonsingular impose extra constraint $\det A=1$ on your matrices to make it unambiguous. Quite often it turns out that it is easier to not do it and still use all coordinates with one extra "scaling degree of freedom". That's because matrices are way easier to work with than some weird "projective matrix spaces".
One example is mean value. If we don't care about the mean value of $n$ numbers, then we only got $n-1$ numbers left to alter. If you have done linear algebra, then you know that $$\sum_{i=1}^n v_i = k$$ is one requirement determining one dimension of the space of $\bf v$. Degrees of freedom is a concept which is a bit similar to that. In statistics one can say that for each moment which is determined or disregarded, one degree of freedom is lost, because then we can only affect a smaller space of numbers.