Matrix Analysis February 17-19, 2014 Outline Introduction to Matrix Analysis • Why matrices, basic definitions/terms • Matrix multiplication • Determinants Larry Caretto • Inverse of a Matrix Mechanical Engineering 309 Numerical Analysis of • Simultaneous linear equations: matrix from and Gaussian elimination solution Engineering Systems – Matrix rank and its role in determining type February 17–19, 2014 of solutions (unique, infinite, none) • Eigenvalues and eigenvectors 2 Why Matrices Matrix Example • Simplified notation for general statements about mathematical or engineering systems • Simple linkage of two springs with • Have multiple components that are spring constant k and displacements uk interrelated at individual points – Elements in a machine structure • Individual loads, Pk, related to individual – Springs, masses and dampers in a displacements, uk, by matrix equation vibrating system shown below. • Matrix notation provides general 푃1 푘1 −푘1 0 푢1 푢 relationships among components P= 푃2 = −푘1 푘1 + 푘2 −푘2 2 = 푲푼 푃 0 −푘 푘 푢3 3 3 2 2 4 Matrix Basics Row and Column Matrices a a a a • Matrices with only one row or only one 11 12 13 1m • Array of a a a a column are called row or column matrices 21 22 23 2m numbers with (sometimes called row or column vectors) a31 a32 a33 a3m n rows and m A – The row number 1 is usually columns dropped in row matrices as is c11 c1 c c • Components the column number 1 in 21 2 an1 an2 an3 anm column matrices are a(row)(column) c c31 c3 r r r r r • Size of matrix (n x m) or (n by m) is 11 12 13 1m number of rows and columns r1 r2 r3 rm cn1 cn 5 6 ME 309 – Numerical Analysis of Engineering Systems 1 Matrix Analysis February 17-19, 2014 More Matrix Basics Diagonal Matrix • Two matrices are equal (e.g., A = B) a1 0 0 0 • The diagonal matrix – If both A and B have the same size (rows 0 a 0 0 A is a square and columns) 2 0 0 a3 0 matrix with nonzero – If each component of A is the same as the A components only corresponding component of B (aij = bij for on the principal all i and j) diagonal • A square matrix has the same number 0 0 0 an of rows and columns • Components of A are 1 i j • A diagonal matrix, D, has all zeros aidij, where dij is the dij except for the principal diagonal Kroenecker delta 0 i j 7 8 Matrix Operations Null (0) and Unit (I) Matrices • Can add or subtract matrices if they are • For any matrix, A, A + 0 = 0 + A = A; the same size IA = AI = A and 0 A = A 0 = 0 – C = A B only valid if A, B, and C have • The unit (or identity) matrix is a square the same size (rows and columns) matrix; the null matrix, which need not – Components of C, cij = aij bij be square, is sometimes written 0(nxm) • Multiplication by a scalar: C = xA 0 0 0 0 1 0 0 0 – C and A have the same size (rows and 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 columns) 0 I – Components of C, cij = xaij – For scalar division, C = A/x, cij = aij/x 0 0 0 0 0 0 0 1 9 10 Transpose of a Matrix Matrix Multiplication Preview • Transpose of A denoted as AT • Not an intuitive operation. Look at two • Reverse rows and columns; for B = AT coordinate transformations as example y a x a x z b y b y – bij = aji 1 11 1 12 2 1 11 1 12 2 – If A is (n x m), B = AT is (m by n) y2 a21x1 a22x2 z2 b21y1 b22y2 • In MATLAB the apostrophe is used to construct the transpose: B = A’ • Substitute equations for y in terms of x 3 14 into equations for z1 and z2 3 12 6 T A A 12 2 z1 b11[a11x1 a12x2 ] b12[a21x1 a22x2 ] 14 2 0 6 0 z2 b21[a11x1 a12x2 ] b22[a21x1 a22x2 ] 11 12 ME 309 – Numerical Analysis of Engineering Systems 2 Matrix Analysis February 17-19, 2014 Matrix Multiplication Preview II Matrix Multiplication Preview III • Rearrange last set of equations to get • Coefficients as matrix components direct transformation from x to z a a b b c c A 11 12 B 11 12 C 11 12 z [b a b a ]x [b a b a ]x c x c x 1 11 11 12 21 1 11 12 12 22 2 11 1 12 2 a21 a22 b21 b22 c21 c22 z2 [b21a11 b22a21]x1 [b21a12 b22a22]x2 c21x1 c22x2 c c b a b a b a b a C 11 12 11 11 12 21 11 12 12 22 c11 [b11a11 b12a21] c12 [b11a12 b12a22] c21 c22 b21a11 b22a21 b21a12 b22a22 c21 [b21a11 b22a21] c22 [b21a12 b22a22] 2 C = AB if cij bik akj (i 1,2; j 1,2) k1 • Generalize this to any matrix size 13 14 Multiplying Matrices General Matrix Multiplication • For general matrix multiplication, C = AB • For matrix multiplication, C = AB – A has n rows and p columns p – A has n rows and p columns p cij aikbkj cij aikbkj – B has p rows and m columns k 1 – B has p rows and m columns k 1 – C has n rows and m columns (i 1,n; j 1,m) – C has n rows and m columns (i 1,n; j 1,m) – A is left; B is right; C is product – A is left; B is right; C is product (AB) is • For C = AB, we get c by adding prod- ij ij • Example 3 4 product of row i ucts of terms in row i of A (left matrix) 3 0 6 A B 1 2 of A times by terms in column j of B (right matrix) 4 2 0 6 1 column j of B • c = a b + a b + a b + a b + … ij i1 1j i2 2j i3 3j i4 4j 3(3) 0(1) 6(6) 3(4) 0(2) 6(1) 27 6 • In general, AB ≠ BA AB 4(3) 2(1) 0(6) 4(4) 2(2) 0(1) 10 12 15 16 Matrix Multiplication Exercise Matrix Multiplication Exercise II • Consider the following matrices • What is the size of C = AB? 1 2 3 0 2 2 0 A 0 1 4 B 1 3 1 0 1 1 0 2 1 3 1 • Can you find AB, BA or both? • We can find AB, because A has three • C = AB has three rows (like A) and four columns and B has three rows columns (like B) • We cannot find BA because B has four • What is c ? columns and A has three rows 11 • Next chart starts the process of finding all • c11 = (1)(0) + (2)(1) + (3)(2) = 8 components of AB 17 18 ME 309 – Numerical Analysis of Engineering Systems 3 Matrix Analysis February 17-19, 2014 Matrix Multiplication Exercise III Matrix Multiplication Exercise IV • Find c11, c12, c13, and c14 in C = AB • Find c21, c22, c23, and c24 in C = AB • c11 = (1)(0) + (2)(1) + (3)(2) = 8 • c21 = (0)(0) + (-1)(1) + (4)(2) = 7 • c12 = (1)(2) + (2)(3) + (3)(1) = 11 • c22 = (0)(2) + (-1)(3) + (4)(1) = 1 • c13 = (1)(-2) + (28)(111) + 9(3)(3-3) = -9 • c23 = (0)(-2) + (-1)(1) + (4)(-3) = -13 8 11 9 3 C • c14 = (1)(0) + (2)(0) + (3)(1) = 3 • c24 = (0)(0C) + 7(-11)(0) 13+ (44)(1) = 4 19 20 Matrix Multiplication Exercise V Matrix Multiplication Results • Find c31, c32, c33, and c34 in C = AB • Solution for matrix product C = AB • c31 = (1)(0) + (1)(1) + (0)(2) = 1 • c32 = (1)(2) + (1)(3) + (0)(1) = 5 • c = (1)(-2) + (1)(1) + (0)(-3) = -1 33 8 11 9 3 • c34 = (1)(0C) + (71)(01) + (130)(14) = 0 1 5 1 0 21 22 MATLAB/Excel Matrices Determinants • Have seen term-by-term operations on • Looks like a matrix but isn’t a matrix arrays,1 A+B,2 3 A-B, A.*B,0 A./B2 and2 0 A.^n • A square array of numbers with a rule A 0 1 4 B 1 3 1 0 for computing a single value for the • Matrix operations are given by following: 1 1 0 array A+B, A-B, A*B, A/B, A2\B 1and A^n3 1 Example at 1 3 1 9 a11 a12 a13 a11 a12 a13 • MATLAB 푨 = 푨. ^ퟐ = 2 4 4 16 right shows Deta a a a a a calculation of 21 22 23 21 22 23 1 3 1 3 7 15 a a a a a a • A^2: 푨ퟐ = = Det(A), the 31 32 33 31 32 33 2 4 2 4 10 22 determinant of a a a a a a a a a • Excel has mmult array function to 11 22 33 21 32 13 31 12 23 3 x 3 matrix A a a a a a a a a a multiply two arrays 11 32 23 21 12 33 31 22 13 23 24 ME 309 – Numerical Analysis of Engineering Systems 4 Matrix Analysis February 17-19, 2014 More Determinants General Rule for Determinants • Useful in obtaining algebraic • Any size determinant can be evaluated expressions for matrix operations, but by any of the following equations not useful for numerical computation n n n n i j i j – Equation for 2 by 2 determinant Det A (1) aijMij (1) aijMij aijAij aijAij i1 j1 i1 j1 a11 a12 a11 a12 Det A Det a11a22 a21a12 a21 a22 a21 a22 • Can pick any row or any column • Choose row or column with most zeros to 2 • An n x n determinant has n Minors, Mij, simplify calculations obtained by deleting row i and column j • Can apply equation recursively; evaluate a 5 i+j • Cofactors, Aij = (-1) Mij used in general x 5 determinant as a sum of 4 x 4 deter- expressions for determinants 25 minants then get 4 x 4’s in terms of 3 x 3’s26 Determinant Behavior Determinant Behavior II • A determinant is zero if any row or any • If one row (or one column) of a column contains all zeros.