§ 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 [a^2 +]
• Let 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 0.
• 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 X.

• 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 X of k^4, with a non-empty open subset U inside it.
• But now, note that the closure of U must lie in X, since X is a closed set, and the closure U of the subset of a closed set must lie in X.
• 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!