COMBINING THE JORDAN CANONICAL FORM AND THE GRAM-SCHMIDT PROCESS
Let Vbe a nonzero finite-dimensional inner product space over the field F(of either the real or the complex numbers). Let be an F-linear transformation that is represented by a matrix Awith respect to some F-basis of V. Suppose that each eigenvalue of T(equivalently, of A) lies in F. A triangulation result states that Vhas an orthonormal basis with respect to which Tis represented by an upper triangular matrix with entries in F; equivalently, that Ais unitarily similar to some upper triangular matrix with entries in F. A very short and transparent proof of this result is given by using two basic tools from a course on matrix theory or linear algebra, namely, the Jordan canonical form and the Gram-Schmidt process.
finite-dimensional vector space, inner product space, linear transformation, matrix, Jordan canonical form, Gram-Schmidt process, orthonormal basis, similarity, unitarily similar, triangulation theorem.