Stat 5101 Lecture Notes - School of Statistics

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236 Stat 5101 (Geyer) Course NotesE.3 Positive Definite MatricesUsing the spectral theorem, we can prove several interesting things aboutpositive definite matrices.Corollary E.5. A real symmetric matrix A is positive semi-definite if and onlyif its spectrum is nonnegative. A real symmetric matrix A is positive definite ifand only if its spectrum is strictly positive.Proof. First suppose that A is positive semi-definite with spectral decomposition(E.7). Let e i denote the vector having elements that are all zero except thei-th, which is one, and define w = Oe i ,so0≤w ′ Aw = e ′ iO ′ ODO ′ Oe i = e ′ iDe i = d ii(E.8)using O ′ O = I. Hence the spectrum is nonnegative.Conversely, suppose the d ii are nonnegative. Then for any vector w definez = O ′ w,sow ′ Aw = w ′ ODO ′ w = z ′ Dz = ∑ i d iiz 2 i ≥ 0Hence A is positive semi-definite.The assertions about positive definiteness are proved in almost the sameway. Suppose that A is positive definite. Since e i is nonzero, w in (E.8) is alsononzero because e i = O ′ w would be zero (and it isn’t) if w were zero. Thus theinequality in (E.8) is actually strict. Hence the spectrum of is strictly positive.Conversely, suppose the d ii are strictly positive. Then for any nonzero vectorw define z = O ′ w as before, and again note that z is nonzero because w = Ozand w is nonzero. Thus w ′ Aw = z ′ Dz > 0, and hence A is positive definite.Corollary E.6. A positive semi-definite matrix is invertible if and only if it ispositive definite.Proof. It is easily verified that the product of diagonal matrices is diagonal andthe diagonal elements of the product are the products of the diagonal elementsof the multiplicands. Thus a diagonal matrix D is invertible if and only if all itsdiagonal elements d ii are nonzero, in which case D −1 is diagonal with diagonalelements 1/d ii .Since O and O ′ in the spectral decomposition (E.7) are invertible, A isinvertible if and only if D is, hence if and only if its spectrum is nonzero, inwhich caseA −1 = OD −1 O ′ .By the preceding corollary the spectrum of a positive semi-definite matrix isnonnegative, hence nonzero if and only if strictly positive, which (again by thepreceding corollary) occurs if and only if the matrix is positive definite.Corollary E.7. Every real symmetric positive semi-definite matrix A has asymmetric square rootA 1/2 = OD 1/2 O ′(E.9)
E.3. POSITIVE DEFINITE MATRICES 237where (E.7) is the spectral decomposition of A and where D 1/2 is defined to bethe diagonal matrix whose diagonal elements are √ d ii , where d ii are the diagonalelements of D.Moreover, A 1/2 is positive definite if and only if A is positive definite.Note that by Corollary E.5 all of the diagonal elements of D are nonnegativeand hence have real square roots.Proof.A 1/2 A 1/2 = OD 1/2 O ′ OD 1/2 O ′ = OD 1/2 D 1/2 O ′ = ODO ′ = Abecause O ′ O = I and D 1/2 D 1/2 = D.From Corollary E.5 we know that A is positive definite if and only if all thed ii are strictly positive. Since (E.9) is the spectral decomposition of A 1/2 ,wesee that A 1/2 is positive definite if and only if all the √ d ii are strictly positive.Clearly d ii > 0 if and only if √ d ii > 0.
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Stat 5101 Lecture NotesCharles J. G
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Contents1 Random Variables and Chan
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CONTENTSv5.1.4 Variance Matrices ..
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CONTENTSviiD Relations Among Brand
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Chapter 1Random Variables andChange
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1.1. RANDOM VARIABLES 3Sometimes it
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1.1. RANDOM VARIABLES 5that is, X i
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1.2. CHANGE OF VARIABLES 7the union
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1.2. CHANGE OF VARIABLES 9Thus even
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1.2. CHANGE OF VARIABLES 11Every in
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1.2. CHANGE OF VARIABLES 13where f
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1.3. RANDOM VECTORS 151.3.1 Discret
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1.4. THE SUPPORT OF A RANDOM VARIAB
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1.5. JOINT AND MARGINAL DISTRIBUTIO
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1.5. JOINT AND MARGINAL DISTRIBUTIO
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1.6. MULTIVARIABLE CHANGE OF VARIAB
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Chapter 2Expectation2.1 Introductio
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2.3. BASIC PROPERTIES 33expectation
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2.3. BASIC PROPERTIES 35that is, th
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2.4. MOMENTS 37The Multiplicativity
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2.4. MOMENTS 39holds for all real-v
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2.4. MOMENTS 41simple, it is often
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2.4. MOMENTS 43In contrast, for all
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2.4. MOMENTS 45is the sum of the ex
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2.4. MOMENTS 47Proof. Just take a i
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2.4. MOMENTS 49What happens to Coro
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2.4. MOMENTS 51This inequality is a
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2.4. MOMENTS 53(a)Why does (2.7) as
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2.5. PROBABILITY THEORY AS LINEAR A
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2.6. PROBABILITY IS A SPECIAL CASE
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2.7. INDEPENDENCE 772.7 Independenc
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2.7. INDEPENDENCE 79Then X and Y ar
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2.7. INDEPENDENCE 81Show that the f
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Chapter 3Conditional Probability an
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3.1. PARAMETRIC FAMILIES OF DISTRIB
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3.2. CONDITIONAL PROBABILITY DISTRI
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3.3. AXIOMS FOR CONDITIONAL EXPECTA
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3.4. JOINT, CONDITIONAL, AND MARGIN
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3.5. CONDITIONAL EXPECTATION AND PR
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Chapter 4Parametric Families ofDist
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4.1. LOCATION-SCALE FAMILIES 113The
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4.2. THE GAMMA DISTRIBUTION 115Show
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4.3. THE BETA DISTRIBUTION 117cours
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4.4. THE POISSON PROCESS 119Hence t
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4.4. THE POISSON PROCESS 121Definit
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4.4. THE POISSON PROCESS 123measure
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4.4. THE POISSON PROCESS 1254-3. A
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Chapter 5Multivariate DistributionT
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5.1. RANDOM VECTORS 129Again like r
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5.1. RANDOM VECTORS 1315.1.6 Covari
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5.1. RANDOM VECTORS 1335.1.7 Linear
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5.1. RANDOM VECTORS 135Example 5.1.
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5.1. RANDOM VECTORS 137Example 5.1.
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5.1. RANDOM VECTORS 139where we hav
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5.2. THE MULTIVARIATE NORMAL DISTRI
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5.3. BERNOULLI RANDOM VECTORS 151Th
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5.3. BERNOULLI RANDOM VECTORS 153yo
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5.4. THE MULTINOMIAL DISTRIBUTION 1
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Chapter 6Convergence Concepts6.1 Un
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6.1. UNIVARIATE THEORY 167It simpli
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6.1. UNIVARIATE THEORY 169density o
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6.1. UNIVARIATE THEORY 171Rewriting
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6.1. UNIVARIATE THEORY 173Comment T
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6.1. UNIVARIATE THEORY 175Problems6
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Chapter 7Sampling Theory7.1 Empiric
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7.1. EMPIRICAL DISTRIBUTIONS 1797.1
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7.1. EMPIRICAL DISTRIBUTIONS 181Def
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7.1. EMPIRICAL DISTRIBUTIONS 183To
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Page 219 and 220: Appendix AGreek LettersTable A.1: T
Page 221 and 222: Appendix BSummary of Brand-NameDist
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Page 237 and 238: Appendix DRelations Among BrandName
Page 239 and 240: Appendix EEigenvalues andEigenvecto
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Page 247: Appendix FNormal Approximations for
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Stat 5101 Lecture Notes - School of Statistics

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