Appendix A Summary of linear algebra

By (1), the map (A, B) → (A · B) is symmetric; by (2), it is bilinear. The inner product is called positive-definite if A · A > 0 for all A =0. In that case the norm of the vector A is |A| ≡ (A · A)1/2. In relativity we deal with inner products that are indefinite: A · A has one sign for some vectors and another for others. In this case the norm, or magnitude, is often defined as |A| ≡ |A · A|1/2. Two vectors A and B are said to be orthogonal if and only if

A · B = 0.

It is often convenient to adopt a set of basis vectors {A1, . . . , An} that are orthonormal: Ai · Aj = 0 if i =j and |Ak| = 1 for all k. This is not necessary, of course. The reader unfamiliar with nonorthogonal bases should try the following. In the two-dimensional Euclidean plane with Cartesian (orthogonal) coordinates x and y and associated Cartesian (orthonormal) basis vectors ex and ey, define A and B to be the vectors A = 5ex + ey, B = 3ey. Express A and B as linear combinations of the nonorthogonal basis {e1 = ex, e2 = ey − ex}. Notice that, although e1 and ex are the same, the 1 and x components of A and B are not the same.

M atrices

A matrix is an array of numbers. We shall only deal with square matrices, e.g.

The dimension of a matrix is the number of its rows (or columns). We denote the elements of a matrix by Aij, where the value of i denotes the row and that of j denotes the column; for a 2 × 2 matrix we have

sions. (Column vectors form a vector space in the usual way.) The following rule governs multiplication of a column vector by a matrix to give a column vector V = A · W:

j=1

Notice that the sum is on the second index of A.

Matrices form a vector space themselves, with addition and multiplication by a number defined by:

A + B = C Cij = Aij + Bij.

aA = B Bij = aAij.

For n × n matrices, the dimension of this vector space is n2. A natural inner product may be defined on this space:

A · B = AijBij. i,j

We can easily show that this is positive-definite. More important than the inner product, however, for our purposes, is matrix multiplication. (A vector space with multiplication is called an algebra, so we are now studying the matrix algebra.) For 2 × 2 matrices, the product is

k=1

Notice that the index summed on is the second of A and the first of B. Multiplication is associative but not commutative; the identity is the matrix whose elements are δij, the Kronecker delta symbol (δij = 1 if i = j, 0 otherwise).

The determinant of a 2 × 2 matrix is

= A11A22 − A12A21.

Given any n × n matrix B and an element Blm (for fixed l and m), we call Slm the (n − 1) × (n − 1) submatrix defined by excluding row l and column m from B, and we call Dlm the determinant of Slm. For example, if B is the 3 × 3 matrix

j=1

In this expression we sum only over j for fixed i. The result is independent of which i was chosen. This enables us to define the determinant of a 3 × 3 matrix in terms of that of a 2 × 2 matrix, and that of a 4 × 4 in terms of 3 × 3, and so on.

Because matrix multiplication is defined, it is possible to define the multiplicative inverse of a matrix, which is usually just called its inverse;

(B−1)ij = (−1)i+jDji/ det (B)

The inverse is defined if and only if det (B) =0.