Matrix Algebra Useful for Statistics,9780470009611

Matrix Algebra Useful for Statistics

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Edition: 1st
ISBN13: 9780470009611
ISBN10: 0470009616
Format: Paperback
Pub. Date: 2006-03-20
Publisher(s): Wiley-Interscience
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Summary

WILEY-INTERSCIENCE PAPERBACK SERIES The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. "This book is intended to teach useful matrix algebra to 'students, teachers, consultants, researchers, and practitioners' in 'statistics and other quantitative methods'.The author concentrates on practical matters, and writes in a friendly and informal style . . . this is a useful and enjoyable book to have at hand." -Biometrics This book is an easy-to-understand guide to matrix algebra and its uses in statistical analysis. The material is presented in an explanatory style rather than the formal theorem-proof format. This self-contained text includes numerous applied illustrations, numerical examples, and exercises.

Author Biography

SHAYLE R. SEARLE, PhD, is Professor Emeritus of Biometry at Cornell University. He is the author of Linear Models, Linear Models for Unbalanced Data, and Generalized, Linear, and Mixed Models (with Charles E. McCulloch), all from Wiley.

Table of Contents

1. Introduction 1(21)
1. The scope of matrix algebra
1(2)
2. General description of a matrix
3(1)
3. Subscript notation
4(2)
4. Summation notation
6(5)
5. Dot notation
11(1)
6. Definition of a matrix
12(3)
7. Vectors and scalars
15(1)
8. General notation
16(1)
9. Illustrative examples
16(1)
Exercises
17(5)
2. Basic Operations 22(38)
1. The transpose of a matrix
22(2)
a. A reflexive operation,
23(1)
b. Vectors,
24(1)
2. Partitioned matrices
24(3)
a. Example,
24(2)
b. General specification,
26(1)
c. Transposing a partitioned matrix,
26(1)
d. Partitioning into vectors,
26(1)
3. The trace of a matrix
27(1)
4. Addition
28(1)
5. Scalar multiplication
29(1)
6. Subtraction
30(1)
7. Equality and the null matrix
31(1)
8. Multiplication
32(18)
a. The inner product of two vectors,
32(1)
b. A matrix–vector product,
33(3)
c. A product of two matrices,
36(2)
d. Existence of matrix products,
38(1)
e. Products with vectors,
39(3)
f. Products with scalars,
42(1)
g. Products with null matrices,
43(1)
h. Products with diagonal matrices,
43(1)
i. Identity matrices,
44(1)
j. The transpose of a product,
44(1)
k. The trace of a product,
45(1)
l. Powers of a matrix,
46(2)
m. Partitioned matrices,
48(1)
n. Hadamard products,
49(1)
9. The Laws of algebra
50(62)
a. Associative laws,
50(1)
b. The distributive law,
51(1)
c. Commutative laws,
51(1)
10. Contrasts with scalar algebra
52(1)
Exercises
53(7)
3. Special Matrices 60(24)
1. Symmetric matrices
60(5)
a. Products of symmetric matrices,
61(1)
b. Properties of AA' and A'A,
61(2)
c. Products of vectors,
63(1)
d. Sums of outer products,
64(1)
e. Elementary vectors,
65(1)
f. Skew-symmetric matrices,
65(1)
2. Matrices having all elements equal
65(3)
3. Idempotent matrices
68(1)
4. Orthogonal matrices
69(4)
a. Definitions,
69(2)
b. Special cases,
71(14)
(i) Helmert matrices,
71(1)
(ii) Givens matrices,
72(1)
(iii) Householder matrices,
72(1)
5. Quadratic forms
73(3)
6. Positive definite matrices
76(2)
Exercises
78(6)
4. Determinants 84(35)
1. Expansion by minors
84(6)
a. First- and second-order determinants,
85(1)
b. Third-order determinants,
86(3)
c. n-order determinants,
89(1)
2. Formal definition
90(2)
3. Basic properties
92(7)
a. Determinant of a transpose,
92(1)
b. Two rows the same,
93(1)
c. Cofactors,
93(2)
d. Adding multiples of a row (column) to a row (column),
95(1)
e. Products,
96(5)
(i) Reduction to triangular form,
96(1)
(ii) Two useful lemmas,
97(1)
(iii) Determinant of a product,
98(1)
4. Elementary row operations
99(4)
a. Factorization,
101(1)
b. A row (column) of zeros,
102(1)
c. Interchanging rows (columns),
102(1)
d. Adding a row to a multiple of a row,
102(1)
5. Examples
103(3)
6. Diagonal expansion
106(3)
7. The Laplace expansion
109(2)
8. Sums and differences of determinants
111(1)
Exercises
112(7)
5. Inverse Matrices 119(36)
1. Introduction: solving equations
119(4)
2. Products equal to I
123(2)
3. Cofactors of a determinant
125(1)
4. Derivation of the inverse
125(4)
5. Conditions for existence of the inverse
129(1)
6. Properties of the inverse
130(1)
7. Some simple special cases
131(2)
a. Inverses of order 2,
131(1)
b. Diagonal matrices,
132(1)
c. I and J matrices,
132(1)
d Orthogonal matrices,
132(1)
e. Idempotent matrices,
133(1)
8. Equations and algebra
133(6)
a. Solving linear equations,
133(3)
(i) Age distributions in wild populations,
133(1)
(ii) Input–output analysis in economics,
134(2)
(iii) Least squares equations,
136(1)
b. Algebraic simplifications,
136(3)
9. Computers and inverses
139(9)
a. The arithmetic of linear equations,
140(2)
b. Rounding error,
142(17)
(i) Addition,
143(1)
(ii) Inverting a matrix,
144(1)
(iii) Solving linear equations,
145(2)
10. Left and right inverses
147(1)
Exercises
148(7)
6. Rank 155(29)
1. Linear combinations of vectors
155(2)
2. Linear transformations
157(2)
3. Linear dependence and independence
159(3)
a. Definitions,
159(2)
b. General characteristics,
161(1)
(i) Sets of vectors,
161(1)
(ii) Some a's zero,
161(1)
(iii) Existence and non-uniqueness of nonzero a's,
161(1)
(iv) Null vectors,
161(1)
4. Linearly dependent vectors
162(4)
a. At least two a's are nonzero,
162(1)
b. Vectors are linear combinations of others,
162(1)
c. Partitioning matrices,
163(1)
d. Zero determinants,
164(1)
e. Inverse matrices,
164(1)
f Testing for dependence (simple cases),
164(2)
5. Linearly independent (LIN) vectors
166(3)
a. Nonzero determinants and inverse matrices,
166(1)
b. Linear combinations of LIN vectors,
167(1)
c. A maximum number of LIN vectors,
167(2)
6. The number of LIN rows and columns in a matrix
169(2)
7. The rank of a matrix
171(1)
8. Rank and inverse matrices
172(1)
9. Permutation matrices
173(2)
10. Full-rank factorization
175(2)
a. Basic development,
175(2)
b. The general case,
177(1)
c. Matrices of full row (column) rank,
177(1)
11. Vector spaces
177(4)
a. Euclidean space,
178(1)
b. Vector spaces,
178(1)
c. Spanning sets and bases,
179(1)
d. Many spaces of order n,
179(1)
e. Subspaces,
180(1)
f. The range and null space of a matrix,
180(1)
Exercises
181(3)
7. Canonical Forms 184(28)
1. Elementary operators
184(2)
a. Row operations,
184(1)
b. Transposes,
185(1)
c. Column operations,
185(1)
d Inverses,
185(1)
2. Rank and the elementary operators
186(1)
a. Rank,
186(1)
b. Products of elementary operators,
186(1)
c. Equivalence,
187(1)
3. Finding the rank of a matrix
187(3)
a. Some special LIN vectors,
187(1)
b. Calculating rank,
188(1)
c. A general procedure,
189(1)
4. Reduction to equivalent canonical form
190(6)
a. Row operations,
190(1)
b. Column operations,
191(1)
c. The equivalent canonical form,
192(2)
d. Non-uniqueness of P and Q,
194(1)
e. Existence is assured
194(1)
f Full-rank factorization,
194(2)
5. Rank of a product matrix
196(3)
6. Symmetric matrices
199(6)
a. Row and column operations,
199(1)
b. The diagonal form,
200(1)
c. The canonical form under congruence,
201(1)
d Two special provisions,
202(3)
(i) Null rows,
202(1)
(ii) Negative elements in the diagonal form,
203(2)
e. Full-rank factorization,
205(1)
7. Non-negative definite matrices
205(4)
a. Diagonal elements and principal minors,
205(1)
b. Congruent canonical form,
206(1)
c. Full-rank factorization,
206(1)
d. Quadratic forms as sums of squares,
206(2)
e. Full row (column) rank matrices,
208(1)
Exercises
209(3)
8. Generalized Inverses 212(15)
1. The Moore—Penrose inverse
212(1)
2. Generalized inverses
212(4)
a. Derivation from row operations,
214(1)
b. Derivation from the diagonal form,
215(1)
3. Other names and symbols
216(1)
4. An algorithm
217(2)
a. An easy form,
217(1)
b. A general form,
217(2)
5. Arbitrariness in a generalized inverse
219(1)
6. Symmetric matrices
220(2)
a. Non-negative definite matrices,
220(1)
b. A general algorithm,
221(1)
c. The matrix X'X,
221(1)
Exercises
222(5)
9. Solving Linear Equations 227(30)
1. Equations having many solutions
227(1)
2. Consistent equations
228(5)
a. Definition,
228(1)
b. Existence of solutions,
229(3)
c. Tests for consistency,
232(1)
3. Equations having one solution
233(2)
4. Deriving solutions using generalized inverses
235(4)
a. Obtaining a solution,
235(1)
b. Obtaining many solutions,
236(1)
c. All possible solutions,
237(2)
d. Combinations of solutions,
239(1)
5. Linearly independent solutions
239(3)
6. An invariance property
242(3)
7. Equations Ax = 0
245(4)
a. General properties,
245(1)
b. Orthogonal solutions,
246(2)
c. Orthogonal vector spaces,
248(1)
8. A complete example
249(2)
9. Least squares equations
251(1)
Exercises
252(5)
10. Partitioned Matrices 257(15)
1. Orthogonal matrices
257(1)
2. Determinants
258(2)
3. Inverses
260(1)
4. Schur complements
261(1)
5. Generalized inverses
261(2)
6. Direct sums
263(2)
7. Direct products
265(2)
Exercises
267(5)
11. Eigenvalues and Eigenvectors 272(48)
1. Introduction: age distribution vectors
272(2)
2. Derivation of eigenvalues
274(2)
3. Elementary properties of eigenvalues
276(3)
a. Eigenvalues of powers of a matrix,
276(1)
b. Eigenvalues of a scalar-by-matrix product,
277(1)
c. Eigenvalues of polynomials,
277(1)
d. The sum and product of eigenvalues,
278(1)
4. Calculating eigenvectors
279(3)
a. A general method,
279(1)
b. Simple roots,
280(1)
c. Multiple roots,
281(1)
5. The similar canonical form
282(8)
a. Derivation,
282(3)
b. Uses,
285(5)
6. Symmetric matrices
290(3)
a. Eigenvalues all real,
290(1)
b. Symmetric matrices are diagonable,
290(1)
c. Eigenvectors are orthogonal,
290(2)
(i) Different eigenvalues,
291(1)
(ii) Multiple eigenvalues,
291(1)
(iii) The canonical form under orthogonal similarity,
291(1)
d. Rank equals number of nonzero eigenvalues,
292(1)
7. Dominant eigenvalues
293(5)
8. Factoring the characteristic equation
298(1)
Exercises
299(6)
11A. Appendix to Chapter 11
305(13)
1. Proving the diagonability theorem
305(3)
a. The number of nonzero eigenvalues never exceeds rank,
305(1)
b. A lower bound on r(A — λkI),
306(1)
c. Proof of the diagonability theorem,
307(1)
d All symmetric matrices are diagonable,
307(1)
2. Other results for symmetric matrices
308(6)
a. Spectral decomposition,
308(1)
b. Non-negative definite (n.n.d.) matrices,
309(3)
c. Simultaneous diagonalization of two symmetric matrices,
312(2)
3. The Cayley—Hamilton theorem
314(2)
4. The singular-value decomposition
316(2)
Exercises
318(2)
12. Miscellanea 320(26)
1. Orthogonal matrices—a summary
320(1)
2. Idempotent matrices—a summary
320(2)
3. The matrix a I + bJ—a summary
322(1)
4. Non-negative definite matrices—a summary
322(1)
5. Canonical forms and other decompositions—a summary
323(2)
6. Matrix Functions
325(1)
a. Functions of matrices,
325(1)
b. Matrices of functions,
325(1)
7. Iterative solution of nonlinear equations
326(1)
8. Vectors of differential operators
327(5)
a. Scalars,
327(1)
b. Vectors,
328(1)
c. Quadratic forms,
329(3)
9. Vec and vech operators
332(28)
a. Definitions,
332(1)
b. Properties of vec,
333(1)
c. Vec-permutation matrices,
334(1)
d Relationships between vec and vech,
334(1)
10. Other calculus results
334(7)
a. Differentiating inverses,
335(1)
b. Differentiating traces,
335(1)
c. Differentiating determinants,
336(2)
d. Jacobians,
338(2)
e. Aitken's integral,
340(1)
f. Hessians,
341(1)
11. Matrices with elements that are complex numbers
341(1)
Exercises
342(4)
13. Applications in Statistics 346(17)
1. Variance—covariance matrices
347(1)
2. Correlation matrices
348(1)
3. Matrices of sums of squares and cross-products
349(3)
a. Data matrices,
349(1)
b. Uncorrected sums of squares and products,
349(1)
c. Means, and the centering matrix,
350(1)
d. Corrected sums of squares and products,
351(1)
4. The multivariate normal distribution
352(3)
5. Quadratic forms and χ²-distributions
355(2)
6. Least squares equations
357(1)
7. Contrasts among means
358(2)
Exercises
360(3)
14. The Matrix Algebra of Regression Analysis 363(29)
1. General description
363(3)
a. Linear models,
364(1)
b. Observations,
364(1)
c. Nonlinear models,
365(1)
2. Estimation
366(2)
3. Several regressor variables
368(1)
4. Deviations from means
369(2)
5. The statistical model
371(1)
6. Unbiasedness and variances
372(1)
7. Predicted y-values
373(1)
8. Estimating the error variance
374(2)
9. Partitioning the total sum of squares
376(2)
10. Multiple correlation
378(1)
11. The no-intercept model
379(1)
12. Analysis of variance
380(2)
13. Testing linear hypotheses
382(4)
a. Stating a hypothesis,
382(1)
b. The F-statistic,
383(2)
c. Equivalent statements of a hypothesis,
385(1)
d. Special cases,
386(9)
(i) H:b= 0,
386(1)
(ii) H:b = b0,
386(1)
(iii) H:λ'b = m,
386(1)
14. Confidence intervals
386(1)
15. Fitting subsets of the x-variables
387(2)
16. Reductions in sums of squares: the R(·|·) notation
389(3)
15. An Introduction to Linear Statistical Models 392(37)
1. General description
392(3)
2. The normal equations
395(2)
a. A general form,
395(1)
b. Many solutions,
396(1)
3. Solving the normal equations
397(2)
a. Generalized inverses of X'X,
397(1)
b. Solutions,
397(2)
4. Expected values and variances
399(1)
5. Predicted y-values
400(1)
6. Estimating the error variance
401(2)
a. Error sum of squares,
401(1)
b. Expected value,
402(1)
c. Estimation,
402(1)
7. Partitioning the total sum of squares
403(1)
8. Coefficient of determination
404(1)
9. Analysis of variance
405(2)
10. The R(·|·) notation
407(1)
11. Estimable functions
408(3)
12. Testing linear hypotheses
411(4)
13. Confidence intervals
415(1)
14. Some particular models
416(8)
a. The one-way classification,
416(2)
b. Two-way classification, no interactions, balanced data,
418(4)
c. Two-way classification, no interactions, unbalanced data,
422(2)
15. The R(·|·) notation (Continued)
424(5)
References 429(4)
Index 433

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