Econ 205 - Slides from Lecture 8Joel SobelSeptember 1, 2010Econ 205 SobelComputational Facts1. det AB = det BA = det A det B2. If D is a diagonal matrix, then det D is equal to the productof its diagonal elements.3. det A is equal to the product of the eigenvalues of A.4. The trace of a square matrix A is equal to the sum of thediagonal elements of A. That is, tr (A) =Pni=1aii. Fact:tr (A) =Pni=1λi, where λiis the ith eigenvalue of A(eigenvalues counted with multiplicity).Econ 205 SobelSymmetric Matrices Have Orthonormal E-vectorsTheoremIf A is symmetric, then we take take the eigenvectors of A to beorthonormal. In this case, the P in the previous theorem has theproperty that P−1= Pt.Econ 205 SobelWhy Eigenvalues?1. They play a role in the study of stability of difference anddifferential equations.2. They make certain computations easy.3. They make it possible to define a sense in which matrices canbe positive and negative that allows us to generalize theone-variable second-order conditions.Econ 205 SobelQuadratic FormsDefinitionA quadratic form in n variables is any function Q : Rn−→ R thatcan be written Q(x) = xtAx where A is a symmetric n × n matrix.When n = 1 a quadratic form is a function of the form ax2. Whenn = 2 it is a function of the forma11x21+ 2a12x1x2+ a22x22(remember a12= a21). When n = 3, it is a function of the forma11x21+ a22x22+ a33x23+ 2a12x1x2+ 2a13x1x3+ 2a23x2x3A quadratic form is second-degree polynomial that has no constantterm.Econ 205 SobelClassificationDefinitionA quadratic form Q(x) is1. positive definite if Q(x) > 0 for all x 6= 0.2. positive semi definite if Q(x) ≥ 0 for all x.3. negative definite if Q(x) < 0 for all x 6= 0.4. negative semi definite if Q(x) ≤ 0 for all x.5. indefinite if there exists x and y such that Q(x) > 0 > Q(y).Econ 205 SobelInterpretationIGeneralized “positivity.”ICheck one-variable case.ILots of indefinite matrices when n > 1.IThink about diagonal matrices for intuition.Q(x) = xtAx =Pni=1aiix2i. This quadratic form is positivedefinite if and only if all of the aii> 0, negative definite if andonly if all of the aii< 0, positive semi definite if and only ifaii≥ 0, for all i negative semi definite if and only if aii≤ 0 forall i, and indefinite if A has both negative and positivediagonal entries.Econ 205 SobelQuadratic Forms and DiagonalizationIAssume A symmetric matrix. It can be written A = RtDR,where D is a diagonal matrix with (real) eigenvalues down thediagonal and R is an orthogonal matrix.IQ(x) = xtAx = xtRtDRx = (Rx)tD (Rx).IThe definiteness of A is equivalent to the definiteness of itsdiagonal matrix of eigenvalues, D.Econ 205 SobelTheorem on DefinitenessTheoremThe quadratic form Q(x) = xtAx is1. positive definite if λi> 0 for all i.2. positive semi definite if λi≥ 0 for all i.3. negative definite if λi< 0 for all i.4. negative semi definite if λi≤ 0 for all i.5. indefinite if there exists j and k such that λj> 0 > λk.Econ 205 SobelComputational TrickDefinitionA principal submatrix of a square matrix A is the matrix obtainedby deleting any k rows and the corresponding k columns. Thedeterminant of a principal submatrix is called the principal minor ofA. The leading principal submatrix of order k of an n × n matrix isobtained by deleting the last n − k rows and column of the matrix.The determinant of a leading principal submatrix is called theleading principal minor of A.Econ 205 SobelDefiniteness TestsTheoremA matrix is1. positive definite if and only if all of its leading principal minorsare positive.2. negative definite if and only if its odd principal minors arenegative and its even principal minors are positive.3. indefinite if one of its kth order leading principal minors isnegative for an even k or if there are two odd leading principalminors that have different signs.The theorem permits you to classify the definiteness of matriceswithout finding eigenvalues.The conditions make sense for diagonal matrices.Econ 205 SobelMultivariable CalculusGoal: Extend the calculus from real-valued functions of a realvariable to general functions from Rnto Rm.Raising the dimension of the range space: easy.Raising the dimension of domain: some new ideas.Econ 205 SobelLinear StructuresIIn R2three linear subspaces: point, lines, entire space.IIn general, we’ll care about 1- and (n − 1)-dimensional subsetsof Rn.Econ 205 SobelAnalytic GeometryDefinitionA line is described by a point x and a direction v. It can berepresented as {z : there exists t ∈ R such that z = x + tv}.If we constrain t ∈ [0, 1] in the definition, then the set is the linesegment connecting x to x + v. Two points still determine a line:The line connecting x to y can be viewed as the line containing xin the direction v. You should check that this is the same as theline through y in the direction v .DefinitionA hyperplane is described by a point x0and a normal directionp ∈ Rn, p 6= 0. It can be represented as {z : p · (z − x0) = 0}. p iscalled the normal direction of the plane.A hyperplane consists of all of the z with the property that thedirection z − x0is normal to p.Econ 205 SobelIn R2lines are hyperplanes. In R3hyperplanes are “ordinary”planes.Lines and hyperplanes are two kinds of “flat” subset of Rn.Lines are subsets of dimension one.Hyperplanes are subsets of dimension n − 1 or co-dimension one.You can have a flat subsets of any dimension less than n.Lines and hyperplanes are not subspaces (because they do notcontain the origin) you obtain these sets by “translating” asubspace that is, by adding the same constant to all of its elements.Econ 205 SobelDefinitionA linear manifold of Rnis a set S such that there is a subspace Von Rnand x0∈ Rnwith S = V + {x0}.In the above definition,V + {x0} ≡ {y : y = v + x0for some v ∈ V }.Officially lines and hyperplanes are linear manifolds and not linearsubspaces.Econ 205 SobelReviewGiven two points x and y, you can construct a line that passesthrough the points.{z : z = x + t(y − x) for some t.}Two-dimensional version:z1= x1+ t(y1− x1) and z2= x2+ t(y2− x2).If you use the equation for z1to solve for t and substitute out youget:z2= x2+(y2− x2) (z1− x1)y1− x1orz2− x2=y2− x2y1− x1(z1− x1) ,which is the standard way to represent the equation of a line (inthe plane) through the point (x1, x2) with slope (y2− x2)(y1− x1).Econ 205 SobelConclusionThe “parametric” representation is essentially equivalent to thestandard representation in