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This expository essay discusses a finite dimensional approach to dilationtheory. How much of dilation theory can be worked out within the realm oflinear algebra? It turns out that some interesting and simple results can beobtained. These results can be used to give very elementary proofs of sharpenedversions of some von Neumann type inequalities, as well as some other strikingconsequences about polynomials and matrices. Exploring the limits of the finitedimensional approach sheds light on the difference between those techniques andphenomena in operator theory that are inherently infinite dimensional, andthose that are not.

Dilation theory in finite dimensions: the possible, the impossible and the unknown