Hermitian, Positive Definite, Unitary, and Normal Matrices

Now consider the set Mn(C) of all n × n matrices with complex entries. If n = 1, we can identify Mn(C) with C. Our goal in this chapter is to build an analogy between C and Mn(C) for n > 1. We will define matrix versions of the concepts of real numbers, positive numbers, pure imaginary numbers, complex numbers of magnitude 1, complex conjugation, the Cartesian decomposition of a complex number, and the polar decomposition of a complex number (among other things). This approach allows us to unify a diverse collection of fundamental results in matrix theory. The analogy with C also provides some motivation for the introduction of certain special classes of matrices (namely unitary, Hermitian, positive definite, and normal matrices) that have many remarkable properties.