In numerical analysis it is often necessary to estimate the condition number CN(T)=‖T‖⋅‖T−1‖ and the norm of the resolvent ‖(ζ−T)−1‖ of a given n×n matrix T . We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the fact that the supremum of CN(T) over all matrices with ‖T‖≤1 and minimal absolute eigenvalue r=minλ∈σ(T)|λ|>0 is the Kronecker bound . This result is subsequently generalized by computing for given ζ in the closed unit disc the supremum of ‖(ζ−T)−1‖, where ‖T‖≤1 and the spectrum 27a620e4816cfc0b00aaeb4" title="Click to view the MathML source">σ(T) of T is constrained to remain at a pseudo-hyperbolic distance of at least r∈(0,1] around ζ . We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occurring Toeplitz matrices are so-called model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space H2 to a finite-dimensional invariant subspace.