GMRES and the minimal polynomial

On Worst-case Gmres, Ideal Gmres, and the Polynomial Numerical Hull of a Jordan Block

When solving a linear algebraic system Ax = b with GMRES, the relative residual norm at each step is bounded from above by the so-called ideal GMRES approximation. This worstcase bound is sharp (i.e. it is attainable by the relative GMRES residual norm) in case of a normal matrix A, but it need not characterize the worst-case GMRES behavior if A is nonnormal. Characterizing the tightness of thi...

The Minimal Polynomial and Some Applications

The easiest matrices to compute with are the diagonal ones. The sum and product of diagonal matrices can be computed componentwise along the main diagonal, and taking powers of a diagonal matrix is simple too. All the complications of matrix operations are gone when working only with diagonal matrices. If a matrix A is not diagonal but can be conjugated to a diagonal matrix, say D := PAP−1 is d...

Minimal Polynomial and Jordan Form

Let V be a vector space over some field k, and let α : V V be a linear map (an ‘endomorphism of V ’). Given any polynomial p with coefficients in k, there is an endomorphism p(α) of V , and we say that p is an annihilating polynomial for α if p(α) = 0. Our first major goal is to see that for any α, the annihilating polynomials can easily be classified: they’re precisely the multiples of a certa...

The Complexity of the Minimal Polynomial

On the Minimal Polynomial of a Matrix