Linear Algebra: Concepts and Methods

A complex matrix is a matrix whose entries are complex numbers. A complex vector space is one for which the scalars are complex numbers. We shall see that many of the results we have established for real matrices and real vector spaces carry over immediately to complex ones, but there are also some significant differences.

In this chapter, we explore these similarities and differences. We look at eigenvalues and eigenvectors of a complex matrix and investigate unitary diagonalisation, the complex analogue of orthogonal diagonalisation. Certain results for real matrices and vector spaces (such as the result that the eigenvalues of a symmetric matrix are real) are easily seen as special cases of their complex counterparts.

We begin with a careful review of complex numbers.

Complex numbers

Consider the two quadratic polynomials, p(x) = x2 – 3x +2 and q(x) = x2 + x + 1. If you sketch the graph of p(x), you will find that the graph intersects the x axis at the two real solutions (or roots) of the equation p(x) = 0, and that the polynomial factorises into two linear factors: p(x) = x2 – 3x + 2 = (x - 1)(x - 2). Sketching the graph of q(x), you will find that it does not intersect the x axis. The equation q(x) = 0 has no solution in the real numbers, and it cannot be factorised over the reals. Such a polynomial is said to be irreducible. In order to solve this equation, we need to use complex numbers.