Theory and Methods of Statistics

Appendix B

Basic Results From Matrix Algebra

This appendix lists some basic definitions, formulas, and results for vectors and matrices which are used in this book. We begin with some simple definitions and elementary results.

B.1 Some Elementary Facts

It is known from the theory of matrices that the number of linearly independent rows of a matrix A equals the number of linearly independent columns, and the rank of A (denoted by rank(A)) is defined to be the number of linearly independent rows of A (or the number of linearly independent columns). For any vector x, x^Tx will be denoted by ∥x∥², which equals the square of the length of x. A matrix A of order n × m is said to have a full rank if $rank(\mathit{A})=min(n,m)$.

For any n × n matrix A, its quadratic ...