Econ 205 - Slides from Lecture 6Joel SobelAugust 30, 2010Econ 205 SobelLinear Algebra BasicsDefinitionn-dimensional Euclidean Space:Rn= R × R × · · · × R × RIf X and Y are sets, thenX × Y ≡ {(x, y) | x ∈ X , y ∈ Y }soRn= {(x1, x2, . . . , xn) | xi∈ R, ∀ i = 1, 2, . . . , n}Econ 205 SobelInterpretations:ICalculus: x ∈ Rnas a list of n real numbers; x = (x1, . . . , xn).ILinear algebra x ∈ Rnis column vector. When we do this wewritex =x1x2...xn.ISometimes an element in Rnis best thought of as a direction.Econ 205 SobelDefinitionThe zero element:0 =00...0Definition (Vector Addition)For x, y ∈ Rnwe havex + y =x1+ y1x2+ y2...xn+ ynVector addition is commutativex + y = y + xEcon 205 SobelDefinition (Scalar Multiplication)For x ∈ Rn, and a ∈ R we haveax =ax1ax2...axnIn other words every element of x gets multiplied by a.Econ 205 SobelVector SpacesDefinition (Vector Space)A vector space is a set V in which the operation of addition andmultiplication by a scalar make sense, and in which operations arecommutative and associative. In addition:0 + v = v (additive identity)for each v there is a −v (additive inverse)A subset of a vector space that is itself a vector space is called asubspace.Euclidean Spaces (Rn) are the leading example of vector spaces.We will need to talk about subsets of Euclidean Spaces that have alinear structure (they contain 0, and if x and y are in the set, thenso is x + y and all scalar multiples of x and y).Econ 205 SobelMatricesDefinitionAn m × n matrix is an element of Mm×nwritten as in the formA =α11α12· · · α1nα21α22· · · α2n.........αm1αm2· · · αmn= [αij]where m denotes the number of rows and n denotes the number ofcolumns.Note An m × n matrix is just of a collection of nm numbersorganized in a particular way. Hence we can think of a matrix asan element of Rm×n. The extra notation Mm×nmakes it possibleto distinguish the way that the numbers are organized.Econ 205 SobelNote Vectors are just a special case of matrices. e.g.x =x1x2...xn∈ Mn×1This notation emphasizes that we think of a vector with ncomponents as a matrix with n rows and 1 column.Econ 205 SobelA2×3=0 1 56 0 2Econ 205 SobelDefinitionThe transpose of a matrix A, is denoted At. To get the transposeof a matrix, we let the first row of the original matrix become thefirst column of the new (transposed) matrix.At=α11α21· · · α1nα12α22· · · α2n.........α1mα2m· · · αnm= [αji]DefinitionA matrix A is symmetric if A = At.So we can see that if A ∈ Mm×n, then At∈ Mn×m.Econ 205 SobelContinuing the example, we see thatAt3×2=0 61 05 2Econ 205 SobelMatrix Algebra[Addition of Matrices] IfAm×n=α11α12· · · α1nα21α22· · · α2n.........αm1αm2· · · αmn= [αij]andBm×n=β11β12· · · β1nβ21β22· · · β2n.........βm1βn2· · · βmn= [βij]thenEcon 205 SobelA + Bm×n= Dm×n=α11+ β11α12+ β12· · · α1n+ β1nα21+ β21α22+ β22· · · α2n+ β2n.........αm1+ βm1αm2+ βm2· · · αmn+ βmn= [δij] = [αij+βij]A + B| {z }m×n=α11+ β11α12+ β12· · · α1n+ β1nα21+ β21α22+ β22· · · α2n+ β2n.........αm1+ βm1αm2+ βm2· · · αmn+ βmn= [αij+βij]Econ 205 SobelDefinition (Multiplication of Matrices)If Am×kand Bk×nare given, then we defineAm×k· Bk×n= Cm×n= [cij]such thatcij≡kXl=1ailbljso note above that the only index being summed over is l.Econ 205 SobelLetA2×3=0 1 56 0 2andB3×2=0 31 02 3ThenA2×3· B3×2|{z }2×2=0 1 56 0 2·0 31 02 3=(0 × 0) + (1 × 1) + (5 × 2), (0 × 3) + (1 × 0) + (5 × 3)(6 × 0) + (0 × 1) + (2 × 2), (6 × 3) + (0 × 0) + (2 × 3)=11 154 24Econ 205 SobelIn general:A · B 6= B · AFor exampleA2×3· B3×46= B3×4· A2×3(The product on the right is not defined.)DefinitionAny matrix which has the same number of rows as columns isknown as a square matrix, and is denoted An×n.Econ 205 SobelDefinitionThere is a special square matrix known as the identity matrix. Anymatrix multiplied by this identity matrix gives back the originalmatrix. The Identity matrix is denoted Inand is equal toInn×n==1 0 . . . 00 1 . . . 0.........0 . . . 0 1.Econ 205 SobelDefinitionA square matrix is called a diagonal matrix if aij= 0 wheneveri 6= j.DefinitionA square matrix is called an upper triangular matrix (resp. lowertriangular if aij= 0 whenever i > j (resp. i < j).Diagonal matrices are easy to deal with. Triangular matrixes arealso somewhat tractable. You’ll see that for many applications youcan replace an arbitrary square matrix with a related diagonalmatrix.Econ 205 SobelFor any matrix Am×nwe have the results thatAm×n· In= Am×nandIm· Am×n= Am×nEcon 205 SobelDefinitionWe say a matrix An×nis invertible or non-singular if ∃ Bn×nsuch thatAn×n· Bn×n| {z }n×n= Bn×n· An×n| {z }n×n= InIf A is invertible, we denote it’s inverse as A−1.So we getA(n×n)· A−1(n×n)| {z }n×n= A−1n×n· An×n| {z }n×n= InA square matrix that is not invertible is called singular.Econ 205 SobelDefinitionThe determinant of a matrix A (written det A = |A|) is definedinductively.n = 1 A(1×1)detA = |A| ≡ a11n ≥ 2 A(n×n)det A =| A |≡a11|A−11| − a12|A−12| + a13|A−13| − · · · ± a1n|A−1n|where A−1jis the matrix formed by deleting the firstrow and jth column of A.Note A−1jis an (n − 1) × (n − 1) dimensional matrix.Econ 205 SobelExamplesIfA2×2= [aij] =a11a12a21a22=⇒ det A = a11a22− a12a21Econ 205 SobelIfA3×3= [aij] =a11a12a13a21a22a23a31a32a33=⇒ det A = a11a22a23a32a33− a12a21a23a31a33+ a13a21a22a31a32Econ 205 SobelDeterminantThe determinant is useful primarily because of the following result:TheoremA matrix is invertible if and only if its determinant 6= 0.Econ 205 SobelDefinitionThe adjoint of a matrix An×n(adj (A)) is the n × n matrix withentry ij equal toadj A = (−1)i+jdet A−ijwhere adj A is the adjoint of A.DefinitionThe Inverse of a matrix An×nis defined asA−1=1det A· adj Awhere adj A is the adjoint of A.Econ 205