Matrix Polynomials, Cayley Hamilton

Cayley Hamilton

The Cayley Hamilton theorem, named after its co-discoverers, states that any given matrix T is a root of its characteristic polynomial. In other words, T is its own eigen value.

Build E by placing the variable x down the main diagonal, and subtracting T. As you recall, the determinant of E gives the characteristic polynomial for T.

Let F be the adjoint of E. Thus E*F gives the determinant, which we will call p(x).

Write F as F₀ + F₁ + F₂ + … F_n, where F_i retains only the coefficients on xⁱ. Thus F₀ is a matrix based on the constants of F, F_n-1 is the identity matrix, and F_n is 0, since no cofactors produce xⁿ.

Now E*F becomes x-T times the sum over F_ixⁱ, while p is the sum of its terms. If c_i is the i^th coefficient on p, we can write c_i = F_i-1-TF_i.

Premultiply the i^th equation by Tⁱ. For example, equation₃ becomes:

T³c₃ = T³F₂ - T⁴F₃

Add these equations together and the right side telescopes to -TⁿF_n, which is 0. The left side becomes p(T). Therefore p(T) = 0, and T is a root of its characteristic polynomial.

Alternate Proof

Similar matrices satisfy the same polynomials, and they have the same eigen values, and the same characteristic polynomials. If T is a root of its own characteristic polynomial, the same holds for every matrix similar to T.

Convert T to jordan form, and find its minimum polynomial m(x), as described in the previous section. This is the lcm of the minimum polynomials of the jordan blocks.

Let a block be diagonal, with eigen value l. Its minimum polynomial is x-l, which is a factor of its characteristic polynomial (x-l)^b, where b is the size of the block.

Next consider a jordan block with l down the diagonal and ones on the subdiagonal. Once again the characteristic polynomial is (x-l)^b. Replace x with the jordan block and evaluate. The expression x-l eaves only the subdiagonal, and when this is raised to the b power, the result is 0.

Each block satisfies its characteristic bpolynomial, and the product of these polynomials gives the characteristic polynomial for T, which is satisfied by all the blocks of T, and by T itself.