Econometrics Exercises 1
MATRIX ALGEBRA

Karim M. Abadir and Jan R. Magnus

Preface [Contents, Photos of Authors]

This volume on matrix algebra and its companion volume on statistics are the first two volumes of the Econometric Exercises series. The two books contain exercises in matrix algebra, probability, and statistics, relating to course material that students are expected to know while enrolled in an (advanced) undergraduate or a postgraduate course in econometrics.

When we started writing this volume, our aim was to provide a collection of interesting exercises with complete and rigorous solutions. In fact, we wrote the book that we — as students — would have liked to have had. Our intention was not to write a textbook, but to supply material that could be used together with a textbook. But when the volume developed we discovered that we did in fact write a textbook, be it one organized in a completely different manner. Thus, we do provide and prove theorems in this volume, because continually referring to other texts seemed undesirable. The volume can thus be used either as a self-contained course in matrix algebra or as a supplementary text.

We have attempted to develop new ideas slowly and carefully. The important ideas are introduced algebraically and sometimes geometrically, but also through examples. It is our experience that most students find it easier to assimilate the material through examples rather than by the theoretical development only.

In proving the more difficult theorems, we have always divided them up in smaller questions, so that the student is encouraged to understand the structure of the proof and will be able to answer at least some of the questions, even if he/she cannot prove the whole theorem. More difficult exercises are marked with an asterisk (*).

One approach to presenting the material is to prove a general result and then obtain a number of special cases. For the student, however, we believe it is more useful (and also closer to scientific development) to first prove a simple case, then a more difficult case, and finally the general result. This means that we sometimes prove the same result two or three times, in increasing complexity, but nevertheless essentially the same. This gives the student who could not solve the simple case a second chance in trying to solve the more general case, after having studied the solution of the simple case.

We have chosen to take real matrices as our unit of operation, although almost all results are equally valid for complex matrices. It was tempting — and possibly would have been more logical and aesthetic — to work with complex matrices throughout. We have resisted this temptation, solely for educational reasons. We emphasize from time to time that results are also valid for complex matrices. Of course, we explicitly need complex matrices in some important cases, most notably in decomposition theorems involving eigenvalues.

Occasionally we have illustrated matrix ideas in a statistical or econometric context, realizing that the student may not yet have studied these concepts. These exercises may be skipped at the first reading.

In contrast to statistics (in particular, probability theory), there only exist a few books of worked exercises in matrix algebra. First, there is Schaum’s outline series with four volumes: Matrices by Ayres (1962), Theory and Problems of Matrix Operations by Bronson (1989), 3000 Solved Problems in Linear Algebra by Lipschutz (1989), and Theory and Problems of Linear Algebra by Lipschutz and Lipson (2001). The only other examples of worked exercises in matrix algebra, as far as we are aware, are Proskuryakov (1978), Prasolov (1994), Zhang (1996, 1999), and Harville (2001).

Matrix algebra is by now an established field. Most of the results in this volume of exercises have been known for decades or longer. Readers wishing to go deeper into the material are advised to consult Mirsky (1955), Gantmacher (1959), Bellman (1970), Hadley (1961), Horn and Johnson (1985, 1991),Magnus (1988), orMagnus and Neudecker (1999), among many other excellent texts.

We are grateful to Josette Janssen at Tilburg University for expert and cheerful typing in LATEX; to Jozef Pijnenburg for constant advice on difficult LATEX questions; to Andrey Vasnev for help with the figures; to Sanne Zwart for editorial assistance; to Bertrand Melenberg, William Mikhail, Maxim Nazarov, Paolo Paruolo, Peter Phillips, Gabriel Talmain, undergraduates at Exeter University, PhD students at the NAKE program in Utrecht and at the European University Institute in Florence, and two anonymous referees, for their constructive comments; and to Scott Parris and his staff at Cambridge University Press for their patience and encouragement. The final version of this book was completed while Jan spent six months as a Jean Monnet Fellow at the European University Institute in Florence.

Updates and corrections of this volume can be obtained from the Econometric Exercises website,

http://us.cambridge.org/economics/ee/econometricexercises.htm

Of course, we welcome comments from our readers.

Karim M. Abadir
Jan R. Magnus