§ Cayley hamilton for 2x2 matrices in sage via AG
- I want to 'implement' the zariski based proof for cayley hamilton in SAGE and show that it works by checking the computations scheme-theoretically.
- Let's work through the proof by hand. Take a 2x2 matrix
[a, b; c, d].
- The charpoly is
|[a-l; b; c; d-l]| = 0, which is
p(l) = (a-l)(d-l) - bc = 0
- This simplified is
p(l) = l^2 - (a + d) l + ad - bc = 0.
- Now, let's plug in
l = [a; b; c; d] to get the matrix eqn
[a;b;c;d]^2 - (a + d)[a;b;c;d] + [ad - bc; 0; 0; ad - bc] = 0.
- The square is going to be
X be the set of
(a, b, c, d) such that the matrices
[a;b;c;d] satisfy their only charpoly.
- Consider the subset
U of the set
(a, b, c, d) such that the matrix
[a;b;c;d] has distinct eigenvalues.
- For any matrix with distinct eigenvalues, it is easy to show that they satisfy their charpoly.
- First see that diagonal matrices satisfy their charpoly by direct computation:
[a;0;0;b] has eigenvalues
(a, b). Charpoly is
l^2 - l(a + b) + ab. Plugging in the matrix, we get
[a^2;0;0;b^2] - [a(a+b);0;0;b(a+b)] + [ab;0;0;ab] which cancels out to
- Then note that similar matrices have equal charpoly, so start with
|(λI - VAV')| = 0. rewrite as
(VλIV' - VAV') = 0, which is
V(λI - A)V' = 0, which is the same
λI - A = 0.
- Thus, this means that a matrix with distinct eigenvalues, which is similar to a diagonal matrix (by change of basis), has a charpoly that satisfies cayley hamilton.
- Thus, the set of matrices with distinct eigenvalues,
U is a subset of
- However, it is not sufficient to show that the system of equations has an infinite set of solutions.
- For example,
xy = 0 has infinite solutions
(x=0, y=k) and
(x=l, y=0), but that does not mean that it is identically zero.
- This is in stark contrast to the 1D case, where a polynomial
p(x) = 0 having infinite zeroes means that it must be the zero polynomial.
- Thus, we are forced to look deeper into the structure of solution sets of polynomials, and we need to come up with the notion of irreducibility.
- See that the space
K^4 is irreducible, where
K is the field from which we draw coefficients for our matrix.
- Next, we note that
X is a closed subset of
k^4 since it's defined by the zero set of the polynomial equations.
- We note that
U is an open subset of
k^4 since it's defined as the non-zero set of the discriminant of the charpoly! (ie, we want non-repeated roots)
- Also note that
U is trivially non-empty, since it has eg. all the diagonal matrices with distinct eigenvalues.
- So we have a closed subset
k^4, with a non-empty open subset
U inside it.
- But now, note that the closure of
U must lie in
X is a closed set, and the closure
U of the subset of a closed set must lie in
- Then see that since the space is irreducible, the closure of
U (an open) must be the whole space.
- This means that all matrices satisfy cayley hamilton!