Suppose you have a squared matrix with rows and columns. Is there a relationship between its eigenvalues and its determinant? And what about its trace?
Determinant of A.
Consider the determinant of , which is a polynomial of degree . The values of that solve the equation , are the eigenvalues of , and are as its degree.
We can write such a polynomial using its roots (its eigenvalues) as follows:
Its constant term is: ,
but it’s also the value of with , which obviously is .
Trace of A.
If we develop the term of degree we obtain:
For simplicity we consider the case of an matrix:
It can be decomposed as follows:
Now we can apply the decomposition on the second row of the second matrix:
And finally we can apply the decomposition on the third row of the last matrix:
The matrices that give a contribute to the degree of are:
So their degree contribute is: .
Generalizing the contribution is: .