Can we reconstruct an $n\times n$ matrix if all we know is the determinant and trace of that matrix (and its size: i.e. what $n$ is)? I would think not, because two scalars wouldn't be enough to tell us about all $n^2$ entries. But then this brings up three questions for me.
- What else do we need to specify a matrix exactly? Obviously, I'm not talking about something for which we can immediately get back the matrix, like for instance, the transpose. But what if we knew all of the eigenvectors and eigenvalues? Or what if we know its Jordan form?
- If all we know is the trace and the determinant, what else can we figure out about the matrix? Obviously we know whether it is invertible or not, but what about say the eigenvalues -- can we figure them out in the $n\ge 3$ case? Can we figure something else out about the matrix?
- Are the trace and the determinant the only invariants of a matrix under change of basis?
Basically, I'm just trying to find out exactly what information is encoded in the trace and determinant of a matrix.
I know that the trace is the sum of the eigenvalues, the determinant is the product of the eigenvalues, and that the determinant of a matrix is the factor by which areas change under the linear transformation $x \mapsto Ax$, where $A$ is the matrix. Is there anything thing else that knowing both of these invariants tell us?