In this post I construct some quantities on symmetric matrices which are invariant under similarity. I don’t know if this is well-known in linear algebra, perhaps Zorn Leung will know about this.
Let be a matrices. The determinant of is defined to be
( is the permutation group of numbers) and its trace is defined to be
It is well-known (especially the former one) that the determinant and the trace of a matrices under similarity. i.e. for any invertible , and . The later identity has an easy proof: first note that , apply this to and , we get .
Several months ago I was thinking about the geometry of rank symmetric matrices and which led me to think about which function “characterizes” those matrices. Consequently, after discussing with Zorn Leung and Bean, I come up with some invariants which are quite interesting themselves.
The idea is actually simple. For a symmetric matrix , it is of rank if and only if the characteristic polynomial has 0 as a root of multiplicity (this is not true in general when is not symmetric, e.g. ) and a polynomial has 0 as a root of multiplicity if and only if . So we are led to find those coefficient of the characteristic polynomial . Two of them are easy to spot: and , i.e.
This can be seen by observing the power of in the expression for example, and also by putting on both sides. Alternatively, it can be seen by the fact that (up to a sign) = sum of roots of the characteristic polynomial = sum of eigenvalues = trace of . Similarly the determinant = product of the eigenvalues = product of the roots of the characteristic polynomial =
What about other terms? It is not difficult either. By observing the power of in the expression , or by differentiating w.r.t. of at , and using the formula etc, we arrive at
where is the muli-index and is the (sub) determinant of the submatrix obtained by deleting all the -th row and -th columns of , . When , we recover to be the original determinant and when , we obtain the trace, since it is the sum of the subdeterminant (i.e. the number itself) along the diagonal.
In particular this proves that
Theorem 1 (Zorn-Bean’s theorem, 2010) A real symmetric matrix is of rank if and only if
where is as defined above.
For example, if , then it is of rank 2 if and only if . It is of rank 1, if and only if and . Finally it is of rank 0 (i.e. ) if and only if and . In the last case, it is easy to see that the last equation forces .