Matrix Algebra Matrix Algebra • Definitions • Addition and Subtraction • Multiplication • Determinant • Inverse • System of Linear Equations • Quadratic Forms • Partitioning • Differentiation and Integration 2.1 Definitions • Scalar (0D matrix) • Vector ( 1D matrix ) Transpose • Matrix ( 2D matrix ) Transpose 2.1 Definitions, contd • Matrix size, [m x n] or [i x j], ( [row x col] ) – Vector, [m x 1] [m x n] = [4 x 2] • Square matrix, (m=n) [m x m] • Symmetric matrix, B=BT 2.1 Definitions, contd • Diagonal matrix • Identity matrix, (unit matrix), AI=A • Zero matrix 2.2 Addition and Subtraction • Vector • Matrix 2.3 Multiplication - scalar • Scalar – vector multiplication • Scalar – matrix multiplication 2.3 Multiplication - vector • Scalar product, (vector – vector multiplication) [1 x n] [n x 1] = [1 x 1] • Length of vector (cf. Pythagoras' theorem) 2.3 Multiplication - vector • Matrix product a1 a1b1 a1b2 a1b3 T ab = a [b b b ]= a b a b a b 2 1 2 3 2 1 2 2 2 3 a3 a3b1 a3b2 a3b3 [m x 1] [1 x n] = [m x n] 2.3 Multiplication - matrix • Matrix – vector multiplication [m x n] [n x 1] = [m x 1] [3 x 2] [2 x 1] = [3 x 1] • Vector – matrix multiplication [1 x m] [m x n] = [1 x n] 2.3 Multiplication - matrix • Matrix – matrix multiplication [m x n] [n x p] = [m x p] Note! 2.3 Multiplication – matrix, contd • Product of transposed matrices • Distribution law 2.4 Determinant • The determinant may be calculated for any square matrix, [n x n] • Cofactor of matrix, A (i=row, k=column) • Expansion formula 2.4 Determinant, examples Cofactors 1 2 Ac = (−1)(1+1) 4 = 4 A = 11 3 4 c (1+2) A12 = (−1) 2 = −2 Determinant det A =1⋅4 + 2(−4) = −2 Cofactors c = − (1+1) ⋅ − ⋅ = − 1 2 3 B11 ( 1) (5 5 6 8) 23 Bc = (−1)(2+1) (2⋅5 − 3⋅8) = −14 B = 4 5 6 21 c (3+1) 7 8 5 B31 = (−1) (2⋅6 − 3⋅5) = −3 Determinant det B =1(−23) + 4⋅14 + 7(−3) =12 2.5 Inverse Matrix • The inverse A-1 of a square matrix A is defined by • The inverse may be determined by the cofactors, where is the adjoint of A and 2.4 Inverse, examples 1 2 4 − 2 A = adjA = 3 4 − 3 1 − −1 4 2 1 What happens if A = adjA / det A = − 3 1 (−2) det A = 0? 1 2 3 − 23 22 − 3 T = B 4 5 6 adjB = −14 −16 6 7 8 5 − 3 6 − 3 − 23 14 − 3 −1 1 B = adjB / det B = − 22 −16 6 12 − 3 6 − 3 2.6 Systems of Linear Equations -number of equations is equal to number of unknowns • Linear equation system A is a square matrix [n x n], x and b [n x 1] vector b = 0 => Homogeneous system b ≠ 0 => Inhomogeneous system • Assume that det( A ) ≠ 0 2.6 Linear Equations -number of equations is equal to number of unknowns • Homogeneous system b = 0, (trivial solution: x = 0) Eigenvalue problems • Inhomogeneous system b ≠ 0 2.8 Quadratic forms and positive definiteness • If • then the matrix A is positive definite • If A is positive definite, all diagonal elements must be positive 2.9 Partitioning The matrix A may be partitioned as and if A may be written 2.9 Partitioning, contd An equation system can be written with introduce The partitioned equation system is written or 2.10 Differentiation and integration • A matrix A • Differentiation • Integration .