p1 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 EG1C2 Engineering Maths : Matrix Algebra Dr Richard Mitchell, Department of Cybernetics AimDescribe matrices and their use in varied applications Syllabus Introduction : why use; definitions & simple processing Determinants and inverses Two Port Networks, for electronics & other systems Gaussian elimination to solve linear equations + Gauss-Jordan Matrix Rank and Cramer's Rule and Theorem Eigenvalues and eigenvectors, applications incl. state space Vectors - and their relationship with matrices. References K.A.Stroud – Engineering Mathematics – Fifth Edn - Palgrave Glyn James - Modern Engineering Mathematics - Addison Wesley Online Notes

p2 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Matrix Algebra - Introduction Simple systems - defined by equations y = f(x): e.g. y = kx Many systems involve many variables: y 1 = k 11 x 1 + k 12 x 2 + k 13 x 3 y 2 = k 21 x 1 + k 21 x 2 + k 23 x 3 y 3 = k 31 x 1 + k 32 x 2 + k 33 x 3 etc. Matrix techniques allow us to represent these by y = k x Bold letters show these are vector or matrix quantities. Why we use matrices Can group related data and process them together. Can use clever techniques to solve problems. Standard matrix manipulation techniques are available. Can use a computer to process the data: e.g. use MATLAB . In course use only 2 or 3 variables, use computers for more.

p3 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Example: Suspended Mass Resolving forces in horizontal and vertical directions: cos (16.26) T 1 = cos (36.87) T 2 sin (16.26) T 1 + sin (36.87) T 2 = 300 {weight of mass} Simplifies to 0.96 T T 2 = 0 and 0.28 T T 2 = 300 T 1 & T 2 are tensions in two wires. (Angles chosen for easy arithmetic) The system can then be written as A.T = Y T, Y are vectors - 1 column 2 rows, A is a matrix - 2 columns 2 rows

p4 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Example : Electronic Circuit Using Kirchhoff’s Voltage and Current Laws First Loop 12 = 18 i i 2 or18 i i 2 = 12 Second Loop 10 i 2 = 15 i 3 or-10 i i 3 = 0 Summing currents i 1 = i 2 + i 3 or-i 1 + i 2 + i 3 = 0

p5 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Matrix Definitions Rectangular array of numbers, complex numbers, functions,.. If r rows & c columns, r*c elements, r*c is order of matrix. Square matrix: a 11, a 22,.. a nn form the main diagonal A vector has one column or one row A is n * m matrix Square if n = m a ij is in row i column j

p6 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Simple Matrix Operations Equality : A and B are identical if, they are of the same size, m = r and n = c, and corresponding elements are same ie a ij = b ij for all i,j Illustrate these by defining A (size m*n) and B (size r*c) A = B, but A  C, A  D

p7 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Matrix Addition A and B must have same size: result is a matrix also of the same size, call it matrix R, in which for all elements, r ij = a ij + b ij. NB A + B = B + A (A + B) + C = A + (B + C) = A + B + C

p8 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Matrix Subtraction A and B must have same size: result is a matrix also of the same size, call it R, in which for all elements, r ij = a ij - b ij. Matrix Scalar Multiplication Each element in the matrix is multiplied by a scalar constant: R = k.A Thus, each r ij =k.a ij. Note, k * (A + B) = k*A + k*B

p9 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Matrix Multiplication R = A*B, number of columns in A = number of rows in B; R has number of rows as A and number of columns as B. e.g; if A is 2 * 3, B is 3 * 4, then A * B is 2 * 4 matrix. Do first element of i th row of A * first element of j th column of B Multiply second, third, etc. elements of these rows and columns Find the sum of each product and store in r ij If A * B ok, then B * A is only possible if A & B are square. A * B  B * A in general. A*(B*C) = (A*B)*C = A*B*C A*(B+C) = (A*B)+(A*C) (k*A)*B=k*(A*B)=A*(k*B) scalar k

p10 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Examples & Exercise

p11 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Multiplication and Example Systems Exercise Express equations 5x + 2y = 16 and 3x = 18 – 4y in matrix form Suspended Mass: A 2*2 matrix times a 2*1 vector = a 2*1 vector Electronic Circuit: A 3*3 matrix times a 3*1 vector = a 3*1 vector

p12 RJM 06/08/02EG1C2 Engineering Maths: Matrix Algebra 1 Matrix Transpose (A transposed is A T ) If R = A T, then r ij = a ji. If A is size m*n, then A T is size n*m. Note: (A T ) T =A (A+B) T =A T +B T (A*B) T =B T *A T (kA) T =kA T If A T =A, A is symmetrix matrix. If A T =-A, A is skew-symmetrix matrix