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EG1C2 Engineering Maths : Matrix Algebra Dr Richard Mitchell, Department of Cybernetics

EG1C2 Engineering Maths : Matrix Algebra Dr Richard Mitchell, Department of Cybernetics. Aim Describe matrices and their use in varied applications Syllabus Introduction : why use; definitions & simple processing Determinants and inverses Two Port Networks, for electronics & other systems

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EG1C2 Engineering Maths : Matrix Algebra Dr Richard Mitchell, Department of Cybernetics

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  1. EG1C2Engineering Maths : Matrix AlgebraDr Richard Mitchell, Department of Cybernetics • Aim Describe 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 http://www.cyber.rdg.ac.uk/people/R.Mitchell/teach.htm EG1C2 Engineering Maths: Matrix Algebra 1

  2. Matrix Algebra - Introduction • Simple systems - defined by equations y = f(x): e.g. y = kx • Many systems involve many variables: • y1 = k11x1 + k12x2 + k13x3 • y2 = k21x1 + k21x2 + k23x3 • y3 = k31x1 + k32x2 + k33x3 etc. • Matrix techniques allow us to represent these by y = kx • 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. EG1C2 Engineering Maths: Matrix Algebra 1

  3. Example: Suspended Mass T1 & T2 are tensions in two wires. (Angles chosen for easy arithmetic) • Resolving forces in horizontal and vertical directions: • cos (16.26) T1 = cos (36.87) T2 • sin (16.26) T1 + sin (36.87) T2 = 300 {weight of mass} • Simplifies to 0.96 T1 - 0.8 T2 = 0 and 0.28 T1 + 0.6 T2 = 300 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 EG1C2 Engineering Maths: Matrix Algebra 1

  4. Example : Electronic Circuit • Using Kirchhoff’s Voltage and Current Laws • First Loop 12 = 18 i1 + 10 i2 or 18 i1 + 10 i2 = 12 • Second Loop 10 i2 = 15 i3 or -10 i2 + 15 i3 = 0 • Summing currents i1 = i2 + i3 or -i1 + i2 + i3 = 0 EG1C2 Engineering Maths: Matrix Algebra 1

  5. Matrix Definitions • Rectangular array of numbers, complex numbers, functions, .. • If r rows & c columns, r*c elements, r*c is order of matrix. A is n * m matrix Square if n = m aij is in row i column j A vector has one column or one row Square matrix: a11, a22, .. ann form the main diagonal EG1C2 Engineering Maths: Matrix Algebra 1

  6. Simple Matrix Operations Illustrate these by defining A (size m*n) and B (size r*c) • Equality : A and B are identical if, • they are of the same size, m = r and n = c, and • corresponding elements are same ie aij = bij for all i,j A = B, but AC,AD EG1C2 Engineering Maths: Matrix Algebra 1

  7. 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, rij = aij + bij. NB A + B = B + A (A + B) + C = A + (B + C) = A + B + C EG1C2 Engineering Maths: Matrix Algebra 1

  8. 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, rij = aij - bij. Matrix Scalar Multiplication • Each element in the matrix is multiplied by a scalar constant: • R = k.A Thus, each rij=k.aij. Note, k * (A + B) = k*A + k*B EG1C2 Engineering Maths: Matrix Algebra 1

  9. 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 ith row of A * first element of jth column of B Multiply second, third, etc. elements of these rows and columns Find the sum of each product and store in rij 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 Matrix Multiplication EG1C2 Engineering Maths: Matrix Algebra 1

  10. Examples & Exercise EG1C2 Engineering Maths: Matrix Algebra 1

  11. Multiplication and Example Systems 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 Exercise Express equations 5x + 2y = 16 and 3x = 18 – 4y in matrix form EG1C2 Engineering Maths: Matrix Algebra 1

  12. Matrix Transpose (A transposed is AT) If R = AT, then rij = aji. If A is size m*n, then AT is size n*m. Note: (AT)T=A (A+B)T=AT+BT (A*B)T=BT*AT (kA)T=kAT If AT=A, A is symmetrix matrix. If AT=-A, A is skew-symmetrix matrix EG1C2 Engineering Maths: Matrix Algebra 1

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