The first volume of the Econometric Exercises Series, Matrix Algebra contains exercises relating to course material in matrix algebra that students are expected to know while enrolled in an (advanced) undegraduate or a postgraduate course in econometrics or statistics. The book features a comprehensive collection of exercises with complete answers. More than just a collection of exercises, the volume is a textbook organized in a completely different manner than the usual textbook. It can be used as a self-contained course in matrix algebra or as a supplementary text.

### Contents

Part I. Vectors: 1. Real vectors; 2 Complex vectors; Part II. Matrices: 3. Real matrices; 4. Complex matrices; Part III. Vector Spaces: 5. Complex and real vector spaces; 6. Inner-product space; 7. Hilbert space; Part IV. Rank, Inverse, and Determinant: 8. Rank; 9. Inverse; 10. Determinant; Part V. Partitioned Matrices: 11. Basic results and multiplication relations; 12. Inverses; 13. Determinants; 14. Rank (in)equalities; 15. The sweep operator; Part VI. Systems of Equations: 16. Elementary matrices; 17. Echelon matrices; 18. Gaussian elimination; 19. Homogeneous equations; 20. Nonhomogeneous equations; Part VII. Eigenvalues, Eigenvectors, and Factorizations: 21. Eigenvalues and eigenvectors; 22. Symmetric matrices; 23. Some results for triangular matrices; 24. Schur's decomposition theorem and its consequences; 25. Jordan's decomposition theorem; 26. Jordan chains and generalized eigenvectors; Part VIII. Positive (Semi)Definite and Idempotent Matrices: 27. Positive (semi)definite matrices; 28. Partitioning and positive (semi)definite matrices; 29. Idempotent matrices; Part IX. Matrix Functions: 30. Simple functions; 31. Jordan representation; 32. Matrix-polynomial representation; Part X. Kronecker Product, Vec-Operator, and Moore-Penrose Inverse: 33. The Kronecker product; 34. The vec-operator; 35. The Moore-Penrose inverse; 36. Linear vector and matrix equations; 37. The generalized inverse; Part XI. Patterned Matrices, Commutation and Duplication Matrix: 38. The commutation matrix; 39. The symmetrizer matrix; 40. The vec-operator and the duplication matrix; 41. Linear structures; Part XII. Matrix Inequalities: 42. Cauchy-Schwarz type inequalities; 43. Positive (semi)definite matrix inequalities; 44. Inequalities derived from the Schur complement; 45. Inequalities concerning eigenvalues; Part XIII. Matrix calculus: 46. Basic properties of differentials; 47. Scalar functions; 48. Vector functions; 49. Matrix functions; 50. The inverse; 51. Exponential and logarithm; 52. The determinant; 53. Jacobians; 54. Sensitivity analysis in regression models; 55. The Hessian matrix; 56. Least squares and best linear unbiased estimation; 57. Maximum likelihood estimation; 58. Inequalities and equalities.

### Review

"This collection of exercises in matrix algebra is useful as a text or a reference. It is clearly written and very thorough in its coverage of results used in econometrics. Besides basic topics, it has an excellent treatment of matrix inequalities, vectorization, and matrix calculus. It belongs on every econometrician's bookshelf." -Peter Schmidt, Michigan State University