Number of independent entries of a matrix

[kent davidge]

Is there an easy way to figure out the number of independent parameters a given matrix has?

For example, a general, real, n x n matrix has n^2 entries and that's easy to realize cause we have a squared array of real numbers. What if this matrix is orthogonal?

 

[RPinPA]

For an orthogonal matrix:
Consider the rows, call them $A_i, i = 1,...,n$. You have the requirement that the rows are normalized, $A_i A_i^T = 1$ for all i. That's $n$ conditions.

And you have that $A_i A_j^T = 0$ for all $i \neq j$. That's $n(n-1)/2$ conditions. So I believe that removes $n + n(n-1)/2$ degrees of freedom from the entries of $A$, leaving $n^2 - n - n(n-1)/2$ $= n(n - 1) - n(n-1)/2$ $= n(n-1)/2$. Of course those are nonlinear constraints so I'm not sure the arithmetic must work out exactly that way.

But assuming that it does, then a 2 x 2 has one degree of freedom and a 3 x 3 has three. Are the most general 2 x 2 and 3 x 3 matrices rotation matrices? I think so, and that would make sense. A 2 x 2 rotation matrix has one parameter, the angle. A 3 x 3 rotation matrix has two angles specifying the direction of the rotation axis, and a third expressing the amount of rotation.

You can definitely do this kind of counting for linear constraints. For instance, requiring that a matrix be symmetric , $a_{ij} = a_{ji}$ for $j \neq i$ means that you have $n^2 - n(n-1)/2$ $= n^2 - (n^2/2) + (n/2)$ $= (n^2/2) + (n/2)$ $= n(n+1)/2$ free parameters, the upper triangle (or lower triangle) of the matrix.

 

[fresh_42]

Is there an easy way to figure out the number of independent parameters a given matrix has?

For example, a general, real, n x n matrix has n^2 entries and that's easy to realize cause we have a squared array of real numbers. What if this matrix is orthogonal?

How many conditions do you get out of $A^\tau\cdot A=I\,?$