Example 2

Let A be a unital algebra of finite rank over K. For $x\in A$ to be invertible, it is necessary and sufficient that the linear mapping $y\mapsto xy$ of A into A should have nonzero determinant. It follows that the spectrum of $x$ is the set of roots of the characteristic polynomial of $x$ (A, III, p. 110). If A is the endomorphism algebra of a finite-dimensional vector space V over K, the spectrum of $x$ is therefore the set of eigenvalues of $x$. This is not always so when V is of infinite dimension (cf. I, p. 153, exercise 2).