Analysis of Financial Time Series, Third Edition

Appendix A: Review of Vectors and Matrices

In this appendix, we briefly review some algebra and properties of vectors and matrices. No proofs are given as they can be found in standard textbooks on matrices (e.g., Graybill, 1969).

An m × n real-valued matrix is an m × n array of real numbers. For example,

is a 2 × 3 matrix. This matrix has two rows and three columns. In general, an m × n matrix is written as

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The positive integers m and n are the row dimension and column dimension of . The real number aij is referred to as the (i, j)th element of . In particular, the elements aii are the diagonal elements of the matrix.

An m × 1 matrix forms an m-dimensional column vector, and a 1 × n matrix is an n-dimensional row vector. In the literature, a vector is often meant to be a column vector. If m = n, then the matrix is a square matrix. If aij = 0 for i ≠ j and m = n, then the matrix is a diagonal matrix. If aij = 0 for i ≠ j and aii = 1 for all i, then is the m × m identity matrix