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MATH 1046 Introduction to Matrices (Sections 3.1 and 3.2) Alex Karassev Matrices • An m x n matrix is a rectangular array of numbers with m rows and n columns j-th column a11 a21 A ai1 am11 a m1 a12 a22 ... ... a1 j a2 j ... ... ai 2 ... aij ... am11 ... am1 j ... am 2 ... am j ... a1n a2 n ain i-th row am1 n am n Some examples • • • • • • Coefficient matrix of a linear system Distance matrix Graph adjacency matrix Matrices in computer graphics Matrices in optimization problems Games Example: distance matrix (in km) North Bay Toronto Ottawa North Bay 0 340 360 Toronto 340 0 450 Ottawa 360 450 0 Example: graph adjacency matrix 1 1 0 0 Timmins 1 North Bay 2 Ottawa 3 Toronto 4 1 1 1 1 0 1 1 1 0 1 1 1 Matrix operations • • • • Multiplication by scalars Addition and subtraction Multiplication Transpose Multiplication by scalars • Example: distance matrices in km and mi km NB T O mi NB T O NB 0 340 360 NB 0 340/1.6 360/1.6 T 340 0 450 T 340/1.6 0 450/1.6 O 360 450 0 O 360/1.6 450/1.6 0 • All entries are multiplied by 1/1.6 Multiplication by scalars a11 a21 A ai1 am 11 a m1 a12 a22 ai 2 am 11 am 2 ca11 ca21 cA cai1 cam 11 ca m1 ca12 ca22 cai 2 cam 11 cam 2 ... ... ... ... ... ... ... ... ... ... a1 j a2 j aij am 1 j am j ca1 j ca2 j caij cam 1 j cam j a1n a2 n ain am 1 n am n ... ... ... ... ... ... ... ... ... ... ca1n ca2 n cain cam 1 n cam n Addition First Apples Basket Second Apples Basket Pears Pears Red 3 5 Red 1 0 Green 4 6 Green 10 3 Total Apples Pears Red 3+1 5+0 Green 4+10 6+3 Addition a11 a A 21 ... a m1 a12 a22 ... am 2 ... a1n ... a2 n ... ... ... amn a11 b11 a b 21 21 A B ... a b m1 m1 b11 b12 ... b1n b ... b b 2n 22 B 21 ... ... ... ... b b ... b mn m1 m 2 a12 b12 a22 b22 ... am 2 bm 2 ... a1n b1n ... a2 n b2 n ... ... ... amn bmn Matrix miultiplication • Example: 2 x 2 linear system a11 x1 a12 x 2 c1 a21 x1 a22 x 2 c 2 a11 a12 a21 a22 • Linear substitution x1 b11 y1 b12 y 2 x2 b21 y1 b22 y 2 b11 b12 b21 b22 x1 b11 y1 b12 y 2 x2 b21 y1 b22 y 2 a11x1a12 x 2 c1 a21x1a22 x 2 c 2 b11 b12 b21 b22 a11 a12 a21 a22 Substitution: a11 (b11 y1b12 y 2 ) a12 (b21 y1 b22 y 2 ) c1 a21 (b11 y1b12 y 2 ) a22 (b21 y1 b22 y 2 ) c 2 What system do we get in terms of y1 and y2? a11 (b11 y1b12 y 2 ) a12 (b21 y1 b22 y 2 ) c1 a21 (b11 y1b12 y 2 ) a22 (b21 y1 b22 y 2 ) c 2 (a11b11 a12b21 ) y1(a11b12 a12b22 ) y 2 c1 (a21b11 a22b21 ) y1(a21b12 a22b22 ) y 2 c 2 New Coefficient Matrix: a11b11 a12b21 a21b11 a22b21 a11b12 a12b22 a21b12 a22b22 Matrix multiplication: 2 x 2 case a11 A a21 a12 a22 a11b11 a12b21 AB a21b11 a22b21 b11 b12 B b21 b22 a11b12 a12b22 a21b12 a22b22 Matrix multiplication: 2 x 2 case a11 A a21 a12 a22 Dot product: a11b11 a12b21 AB a21b11 a22b21 b11 b12 B b21 b22 a11b12 a12b22 a21b12 a22b22 Matrix multiplication: general case • Let A = (aij) be m x n matrix and B be n x k matrix • The product AB = C = (cij) is an m x k matrix defined as follows cij = ai1 b1j + ai2 b2j+ ci3 b3j+…+ain bnj • Note: cij is the dot product of i-th row of A and j-th column of B Is it possible that AB≠BA? Is it possible that AB≠BA? Yes! 0 1 A 1 0 1 1 B 0 1 0 1 AB 1 1 1 1 BA 1 0 Square matrix • A matrix is called square if m=n, i.e. the number of rows is the same as the number of columns • A square matrix has the diagonal: all entries of the form aii a11 a21 a 31 a12 a22 a32 a13 a23 a33 Zero Matrix and Identity Matrix • Zero matrix: an n x n matrix such that all entries are zeros • Identity matrix: an n x n matrix such that all diagonal entries are 1, all other entries are 0 0 0 0 O 0 0 0 0 0 0 1 0 0 I 0 1 0 0 0 1 Properties: addition and multiplication by scalars • Addition of matrices and multiplication by scalars have the same properties as in the case of vectors or real numbers: • (A+B)+C=A+(B+C) • A+B = B+A • A+O = A • If -A = (-1)+A then A+(-A) = O • (cd)A= c(dA) • 1A = A • (c+d)A = cA + dA • c(A+B) = cA +cB Properties: matrix multiplication • Assuming the dimension of matrices allow to perform the operations, we have the following: • • • • • • (AB)C=A(BC) A(B+C)= AB+AC (B+C)A= BA+CA IA = A I = A OA = AO = O (cA)B= A (cB) = c(AB) Powers • For a square matrix A, define Ak = A A … A (product of A with itself k times) • Question: Is it possible to have A2 = O for a non-zero A? Yes! 0 1 A 0 0 Scalar matrices • Matrix A is called scalar if A = aI for some real number a • Exercise a) Prove that for a scalar n x n matrix A and for any n x n matrix B we have AB= BA = aB b) Prove (at least for 2 x 2 case) that in fact for an arbitrary square matrix A we have AB= BA for any B if and only if A is scalar Matrix Transpose • AT = B such that bij = aji • If A is m x n matrix AT is n x m matrix • If v is a row vector vT is a column vector, and vice versa • In general: columns of AT are rows of A rows of AT are columns of A • For a square matrix A this can be viewed as a flipping with respect to the diagonal: a11 A a21 a 31 a12 a22 a32 a13 a23 a33 a11 T A a12 a 13 a21 a22 a23 a31 a32 a33 Properties of transpose operation • • • • • (AT)T = ? (cA) T= ? (A+B)T = ? (AB)T = ? (Ak)T= ? Properties of transpose operation • • • • • (AT)T = A (cA) T=cAT (A+B)T = AT+ BT (AB)T =BTAT (Ak)T= (AT) k • Note: in general AAT ≠ATA (exercise: find an example!); square matrices that commute with its transpose are called normal Symmetric and skew-symmetric matrices • A square matrix is called Symmetric if AT = A • Examples: distance matrix, adjacency matrix Skew-symmetric (or antisymmetric) if AT = -A • Examples: matrices of some games Symmetric and Skew-symmetric matrices • (A+AT)T= AT + A = A + AT • (AAT)T=(AT)TAT =AAT A+AT and AAT are symmetric for any square A • (A-AT)T=AT - A = -(A - AT) A-AT is skew-symmetric for any square A • Note: for any square matrix A we have A=1/2(A+AT) + 1/2 (A-AT) Symmetric matrices: Exercise • Give an example of two symmetric 2x2 matrices whose product is not symmetric • Prove that the product of two symmetric square matrices A and B is symmetric if and only if AB = BA