Preliminary Course on Mathematics winter term 2014/2015 Veronika - PowerPoint PPT Presentation

Preliminary Course on Mathematics – winter term 2014/2015 Veronika Penner vpenner@economics.uni-kiel.de CAU Kiel

Literature • Parts of the course content are based on ‘ A Cook-Book Of Mathematics ’ by V. Vinogradov (Publisher: CERGE-EI 1999) Download link: http://www.cerge-ei.cz/pdf/lecture_notes/LN01.pdf ▫ recipes for solving problems students might face in their studies of economics ▫ target audience is supposed to have some mathematical background (BA level) ▫ the main goal is to refresh students' knowledge of mathematics ▫ supplementary reading list is provided • Further literature ▫ K. Sydsaeter, P. J. Hammond and A. Strom - Essential Mathematics for Economic Analysis, Prentice Hall International, 2012 C.P. Simon and L. Blume - Mathematics for Economists, Norton, 1994 ▫ → C.P Simon and L. Blume - Answers Pamphlet for Mathematics for Economists available online

Part 1 – Linear Algebra Veronika Penner vpenner@economics.uni-kiel.de CAU Kiel

Outline Part 1 – Linear algebra I. Vector and matrix algebra III. Eigenvalue problems ▫ Operations ▫ Linear transformations ▫ Scalar product ▫ Eigenvalues ▫ Special matrices ▫ Hessian matrix ▫ Determinant ▫ Matrix inverse II. Systems of equations ▫ Linear combinations ▫ Rank and basis ▫ Solution methods ▫ Nonlinear equations

Vector and matrix algebra

Definition 𝑏 11 ⋯ 𝑏 1𝑜 • Row/column order of matrices ⋮ ⋱ ⋮ 𝐵 𝑛𝑜 = ▫ rows in the first index, 𝑏 𝑛1 … 𝑏 𝑛𝑜 columns in the second 𝐵 𝑛𝑜 = 𝑏 𝑗𝑘 [𝑛𝑦𝑜] • Notation • Dimensions (or order) 𝐵 𝑜 = 𝑏 𝑗𝑗 [𝑜𝑦𝑜] ▫ square ▫ special cases of matrices → vectors 𝑏 11 ⋮ 𝐵 𝑛1 = 𝑏 𝑗1 [𝑛𝑦1] =  column and row vectors 𝑏 𝑛1 𝐵 1𝑜 = 𝑏 1𝑗 [1𝑦𝑜] = (𝑏 11 , ⋯ , 𝑏 1𝑜 )

Matrix operations • Addition/Substraction ▫ only if the matrices have the 𝐵 𝑛𝑜 ±𝐶 𝑛𝑜 = 𝑏 𝑗𝑘 ± 𝑐 𝑗𝑘 [𝑛𝑦𝑜] same dimension • Scalar multiplication 𝜇𝐵 𝑛𝑜 = 𝜇𝑏 𝑗𝑘 [𝑛𝑦𝑜] • Multiplication 𝐵 𝑛𝑜 ∙ 𝐶 𝑜𝑙 = 𝑑 𝑗𝑘 𝑛𝑦𝑙 ▫ note the dimensions! ▫ inner dimensions must be the 𝑥ℎ𝑓𝑠𝑓 𝐷 𝑛𝑙 = 𝑑 𝑗𝑘 [𝑛𝑦𝑙] same 𝑜 and 𝑑 𝑗𝑘 = 𝑏 𝑗𝑚 𝑐 𝑚𝑘 𝑚=1

Examples 9 10 𝐵 23 = 3 2 1 6 , 𝐶 32 = 12 11 • Example 1 4 5 14 13 ⇒ 𝐵 23 ∙ 𝐶 32 = 3 ∙ 9 + 2 ∙ 12 + 1 ∙ 14 3 ∙ 10 + 2 ∙ 11 + 1 ∙ 13 4 ∙ 10 + 5 ∙ 11 + 6 ∙ 13 4 ∙ 9 + 5 ∙ 12 + 6 ∙ 14 = 27 + 24 + 14 30 + 22 + 13 40 + 55 + 78 36 + 60 + 84 65 65 = 173 = 𝐷 22 , 𝐷 ∈ 𝕅 2×2 180 1 • Example 2 𝑣 = 2,1,0 , 𝑣 ∈ 𝕅 1×3 𝑤 = , 𝑣 ∈ 𝕅 3×1 2 1 1 1. 𝑣 ∙ 𝑤 = 2,1,0 ∙ = 4 2 1 1 2 1 0 2. 𝑤 ∙ 𝑣 = ∙ 2,1,0 = ∈ 𝕅 3×3 2 4 2 0 1 2 1 0

Laws of operations 𝐵 + 𝐶 = 𝐶 + 𝐵 • Commutative/associative law 𝐵 + 𝐶 + 𝐷 = 𝐵 + (𝐶 + 𝐷) of addition • Associative law of 𝐵(𝐶𝐷) = (𝐵𝐶)𝐷 multiplication 𝐵 𝐶 + 𝐷 = 𝐵𝐶 + 𝐵𝐷 • Distributiv law: 𝐶 + 𝐷 𝐵 = 𝐶𝐵 + 𝐷𝐵 • NO commutative law of 𝐵𝐶 ≠ 𝐶𝐵 multiplication 𝐵 = 2 0 8 , 𝐶 = 7 2 3 3 6 𝐵𝐶 = 14 30 ≠ 𝐶𝐵 = 20 4 16 24 69 21

Scalar product • Geometric interpretation: Product of the magnitudes of the two vectors and the cosine of the angle between them 𝑏 ∙ 𝑐 = 𝑏 𝑐 cos (𝜄) • If cos 𝜄 = 90° then 𝑏 ∙ 𝑐 = 0 , a and b are orthogonal • Algebraic interpretation: Sum of the products of the corresponding entries of the two vectors 𝑏 1 𝑐 1 𝑜 ⋮ ⋮ 𝑏 = , 𝑐 = → 𝑏 ∙ 𝑐 = 𝑏 𝑗 𝑐 𝑗 𝑏 𝑜 𝑗=1 𝑐 𝑜 1 2 𝑏 = 3 , 𝑐 = 4 → 𝑏 ∙ 𝑐 = 1 ∙ 2 + 3 ∙ 4 + 5 ∙ 6 = 44 5 6

Special matrices 1 ⋯ 0 𝐵I = IA = A • Identity matrices are always ⋮ ⋱ ⋮ 𝐽 𝑜 = square matrices 𝐵IB = AB 0 ⋯ 1 𝑜 0 ⋯ 0 𝐵 + 𝟏 = A • Null matrices can have any dimension 𝟏 𝑜 = ⋮ ⋱ ⋮ 𝐵 + (−A) = 𝟏 0 ⋯ 0 𝑏 11 ⋯ 0 • Diagonal matrices are always square matrices ⋮ ⋱ ⋮ 𝐵 𝑜 = 0 ⋯ 𝑏 𝑜𝑜 𝑏 11 ⋯ 0 • Triangular matrices are always square matrices ⋮ ⋱ ⋮ 𝐵 𝑜 = 𝑏 𝑜1 ⋯ 𝑏 𝑜𝑜

Transpose matrices I 𝐵 𝑈 = 𝑐 𝑗𝑘 [𝑜𝑦𝑛] 𝐵 = 𝑏 𝑗𝑘 [𝑛𝑦𝑜] • Definition 𝑏 𝑘𝑗 = 𝑐 𝑗𝑘 for 𝑗 = 1 … 𝑜 and 𝑘 = 1 … 𝑛 3 4 𝐵 = 3 2 1 𝐵 𝑈 = • Example 6 2 5 4 5 1 6 • Properties ▫ (𝐵 𝑈 ) 𝑈 = 𝐵 ▫ (𝛽𝐵) 𝑈 = 𝛽𝐵 𝑈 ▫ (𝐵 + 𝐶) 𝑈 = 𝐵 𝑈 + 𝐶 𝑈 ▫ (𝐵𝐶) 𝑈 = 𝐶 𝑈 𝐵 𝑈

Transpose matrices II • Transpose matrices are 𝐵 𝑈 = 𝐵 ▫ symmetric if 𝐵 𝑈 = −𝐵 ▫ anti-symmetric if ▫ orthogonal if 𝐵 𝑈 𝐵 = 𝐽 𝐵 𝑈 = 𝐵 𝐵𝐵 = 𝐵 ▫ idempotent if and

• Example 𝑨(𝑦, 𝑧) = 𝑦 2 𝑧 + 6𝑦 − 𝑧 3 • The symmetric Hessian matrix of second partial derivatives(see calculus[Part 2]) is defined as ▫ 𝑨 𝑦 = 2𝑦𝑧 + 6 ⇒ 𝑨 𝑦𝑦 = 2𝑧 ⇒ 𝑨 𝑦𝑧 = 2𝑦 𝜖 2 𝑔 𝜖 2 𝑔 … ▫ 𝑨 𝑧 = 𝑦 2 − 3𝑧 2 𝜖𝑦 1 𝜖𝑦 1 𝜖𝑦 1 𝜖𝑦 𝑜 𝐼 = ⋮ ⋱ ⋮ ⇒ 𝑨 𝑧𝑧 = −6𝑧 𝜖 2 𝑔 𝜖 2 𝑔 ⇒ 𝑨 𝑧𝑦 = 2𝑦 … 𝜖𝑦 𝑜 𝜖𝑦 1 𝜖𝑦 𝑜 𝜖𝑦 𝑜 𝑨 𝑦𝑦 𝑨 𝑦𝑧 𝑨 𝑧𝑧 = 2𝑧 2𝑦 • 𝐼 = −6𝑧 𝑨 𝑧𝑦 2𝑦

Determinant I • Properties ▫ det 𝐵 ∙ 𝐶 = det 𝐵 ∙ det 𝐶 ▫ in general: det 𝐵 + 𝐶 ≠ det 𝐵 + det 𝐶 • Method 1 : (𝐵 2 ) = 𝑏 11 𝑏 12 Determinant of a 2 × 2 -matrix: det 𝑏 22 = 𝑏 11 𝑏 22 − 𝑏 12 𝑏 21 𝑏 21 • Method 2: SARRUS Determinant of a 3 × 3 -matrix 𝑏 11 𝑏 12 𝑏 13 𝑏 11 𝑏 12 𝑏 21 𝑏 22 𝑏 23 𝑏 21 𝑏 22 det (𝐵 3 ) = = 𝑏 11 𝑏 22 𝑏 33 + 𝑏 12 𝑏 23 𝑏 31 + 𝑏 13 𝑏 21 𝑏 32 𝑏 31 𝑏 32 𝑏 33 𝑏 31 𝑏 32 − 𝑏 13 𝑏 22 𝑏 31 − 𝑏 11 𝑏 23 𝑏 32 − 𝑏 12 𝑏 21 𝑏 33

Determinant II 𝑜 (−1) 𝑚+𝑙 ∙ 𝑏 𝑚𝑙 ∙ det det 𝐵 = (𝑁 𝑚𝑙 ) • Method 3: Laplace Expansion 𝑙=1 for some integer 𝑚 𝑗𝑜 1 ≤ 𝑚 ≤ 𝑜 ▫ important concepts  the minor 𝑁 𝑗𝑘 of element 𝑏 𝑗𝑘 is the 2 3 0 𝑁 23 = 2 3 𝐵 = 1 4 5 2 resulting matrix after removing 7 7 2 1 the ith row and the jth col  the co-factor 𝐷 𝑗𝑘 of element 𝑏 𝑗𝑘 : 𝐷 23 = (−1) 2+3 ∙ 5 ∙ −17 = 85 (−1) 𝑚+𝑙 ∙ det (𝑁 𝑚𝑙 ) det 𝐵 = 𝐷 13 + 𝐷 23 + 𝐷 33

Determinant III 𝑜 • Laplace Expansion (−1) 𝑚+𝑙 ∙ 𝑏 𝑚𝑙 ∙ det det 𝐵 = (𝑁 𝑚𝑙 ) 𝑙=1 for some integer 𝑚 𝑗𝑜 1 ≤ 𝑚 ≤ 𝑜 ▫ examples 2 3 0 𝐵 = 2 3 𝐵 = 1 4 5 2 7 7 2 1 𝐷 21 = (−1) 2+1 ∙ 1 ∙ 3 = −3 𝐷 21 = (−1) 2+1 ∙ 7 ∙ 3 = −21 𝐷 22 = (−1) 2+2 ∙ 4 ∙ 2 = 8 𝐷 22 = (−1) 2+2 ∙ 2 ∙ 2 = 4 𝐷 23 = (−1) 2+3 ∙ 5 ∙ (−17) = 85 det 𝐵 = 𝐷 21 + 𝐷 22 + 𝐷 23 = 90 det 𝐵 = 𝐷 21 + 𝐷 22 = − 17

Determinant IV • Simplification rules for calculation 1 – the multiplication of any row/column by a scalar changes the determinant according to the scalar 2 – the interchange of two rows/columns changes the sign of the determinant 3 – the determinant is not affected by linear transformations of rows/columns 4 – if two rows/columns are identical, the determinant equals zero 5 – the determinant of a diagonal/triangular matrix is the product of its diagonal elements

Determinant V 4 −1 6 −2 2 • Example 0 5 −3 𝐵 = 2 −1 0 8 −2 0 0 2 2 −1 −1 3 1 0 5 −3 1) divide the first row by 2 (rule 1) det 𝐵 = 2 ∙ 𝑒𝑓𝑢 2 −1 0 8 −2 0 0 2 2 −1 3 −1 1 0 5 −3 det 𝐵 = 2 ∙ 𝑒𝑓𝑢 2) subtract the first row from the 0 0 0 5 −3 third row (rule 3) 0 0 2 det 𝐵 = 2 ∙ 2 ∙ −1 ∙ 5 ∙ 2 = −40 3) rule 5

Matrix inverse I • Only non-singular matrices are invertible • Equivalent statements matrix A is invertible ⇔ det 𝐵 ≠ 0 ⇔ rank(A) = order(A) (definition later) ⇔ matrix A is non-singular Several ways to check for singularity

Matrix inverse II 2 3 0 • Method 1: Adjoint matrix 𝐵 = → det 𝐵 = 8 + 105 − 20 − 3 1 4 5 7 2 1 ▫ first check for singularity = 90 ≠ 0 1 𝐵 −1 = 𝑒 𝑗𝑘 , (𝐵) (−1) 𝑗+𝑘 det 𝑒 𝑗𝑘 = (𝑁 𝑘𝑗 ) det ↑ ! 𝑒 11 = 1 𝑒 32 = 1 90 −1 1+1 4 − 10 = −6/90 90 −1 3+2 4 − 21 = 17/90 𝑒 21 = 1 𝑒 13 = 1 90 −1 2+1 1 − 35 = 34/90 90 −1 1+3 15 − 0 = 15/90 𝑒 23 = 1 𝑒 31 = 1 90 −1 2+3 10 − 0 = −10/90 90 −1 3+1 2 − 28 = −26/90 𝑒 33 = 1 𝑒 12 = 1 90 −1 3+3 8 − 3 = 5/90 90 −1 1+2 3 − 0 = −3/90 −6/90 −3/90 15/90 𝑒 22 = 1 90 −1 2+2 2 − 0 = 2/90 𝐵 −1 = 34/90 2/90 −10/90 −26/90 17/90 5/90