Lesson 5: Matrix Algebra (slides)

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Lesson 5Matrix Algebra and The TransposeMath 20September 28, 2007AnnouncementsThomas Schelling at IOP (79 JFK Street), Wednesday 6pmProblem Set 2 is on the course web site. Due October 3Problem Sessions: Sundays 6–7 (SC 221), Tuesdays 1–2 (SC116)My office hours: Mondays 1–2, Tuesdays 3–4, Wednesdays1–3 (SC 323)We do not define A + B if A and B do not have the samedimension.Remember the definition of matrix additionDefinitionLet A = (aij )and B = (bbe matrices. The sum of Am×nij )m×nand B is the matrix C = (cij )defined bym×ncij = aij + bijThat is, C is obtained by adding corresponding elements of A AndB.Remember the definition of matrix additionDefinitionLet A = (aij )and B = (bbe matrices. The sum of Am×nij )m×nand B is the matrix C = (cij )defined bym×ncij = aij + bijThat is, C is obtained by adding corresponding elements of A AndB.We do not define A + B if A and B do not have the samedimension.(so addition is commutative)(b) A + (B + C) = (A + B) + C (so addition is associative)(c) There is a unique m × n matrix O such thatA + O = Afor any m × n matrix A. The matrix O is called the m × nadditive identity matrix.(d) For each m × n matrix A, there is a unique m × n matrix Dsuch thatA + D = O(We write D = −A.) So additive inverses exist.Properties of Matrix AdditionRulesLet A, B, C, and D be m × n matrices.(a) A + B = B + A(b) A + (B + C) = (A + B) + C (so addition is associative)(c) There is a unique m × n matrix O such thatA + O = Afor any m × n matrix A. The matrix O is called the m × nadditive identity matrix.(d) For each m × n matrix A, there is a unique m × n matrix Dsuch thatA + D = O(We write D = −A.) So additive inverses exist.Properties of Matrix AdditionRulesLet A, B, C, and D be m × n matrices.(a) A + B = B + A (so addition is commutative)(so addition is associative)(c) There is a unique m × n matrix O such thatA + O = Afor any m × n matrix A. The matrix O is called the m × nadditive identity matrix.(d) For each m × n matrix A, there is a unique m × n matrix Dsuch thatA + D = O(We write D = −A.) So additive inverses exist.Properties of Matrix AdditionRulesLet A, B, C, and D be m × n matrices.(a) A + B = B + A (so addition is commutative)(b) A + (B + C) = (A + B) + C(c) There is a unique m × n matrix O such thatA + O = Afor any m × n matrix A. The matrix O is called the m × nadditive identity matrix.(d) For each m × n matrix A, there is a unique m × n matrix Dsuch thatA + D = O(We write D = −A.) So additive inverses exist.Properties of Matrix AdditionRulesLet A, B, C, and D be m × n matrices.(a) A + B = B + A (so addition is commutative)(b) A + (B + C) = (A + B) + C (so addition is associative)Properties of Matrix AdditionRulesLet A, B, C, and D be m × n matrices.(a) A + B = B + A (so addition is commutative)(b) A + (B + C) = (A + B) + C (so addition is associative)(c) There is a unique m × n matrix O such thatA + O = Afor any m × n matrix A. The matrix O is called the m × nadditive identity matrix.(d) For each m × n matrix A, there is a unique m × n matrix Dsuch thatA + D = O(We write D = −A.) So additive inverses exist.Well,[A + (B + C)] = aijij + [B + C]ij = aij + (bij + cij )= (aij + bij ) + cijas real numbers= [A + B]ij + cij = [(A + B) + C] .ijProofProof.Let’s prove (b). We need to show that every entry of A + (B + C)is equal to the corresponding entry of (A + B) + C.ProofProof.Let’s prove (b). We need to show that every entry of A + (B + C)is equal to the corresponding entry of (A + B) + C. Well,[A + (B + C)] = aijij + [B + C]ij = aij + (bij + cij )= (aij + bij ) + cijas real numbers= [A + B]ij + cij = [(A + B) + C] .ijDocument Outline
  • Announcements
  • Properties of Matrix Addition
  • Properties of Matrix Multiplication
  • The Transpose
  • Useful Facts