Semi-Riemannian Geometry

Chapter 2 Matrices and Determinants

In this chapter, we review some of the basic results from the theory of matrices and determinants.

2.1 Matrices

Let us denote by Mat _m × n the set of m × n matrices (that is, m rows and n columns) with real entries. When m = n , we say that the matrices are square. It is easily shown that with the usual matrix addition and scalar multiplication, Mat _m × n is a vector space, and that with the usual matrix multiplication, Mat _m × m is a ring.

Let P be a matrix in Mat _m × n , with

The transpose of P is the matrix P ^T in Mat _n × m defined by

The row matrices of P are

and the column matrices of P are

Example 2.1.1

For

the transpose is

and the column matrices are

Theorem 2.1.2

If P and ...