Examples and MatLab. Vector and Matrix Material. Matrix Addition R = A + B. Matrix Equality A = B. Matrix Multiplication R = A * B.

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1 Vector and Matrix Material Examples and MatLab Matrix = Rectangular array of numbers, complex numbers or functions If r rows & c columns, r*c elements, r*c is order of matrix. A is n * m matrix Square if n = m a ij is in row i column j A vector has one row or one column For a square matrix: a 11, a 22,.. a nn form the main diagonal MATLAB is interactive Matrix Laboratory Package Matrices can be defined, for instance, at the prompt. >> A = [ ; ] A= We will now consider some simple matrix operations 20/09/2013 SEMMA13 Eng Maths and Stats 1 20/09/2013 SEMMA13 Eng Maths and Stats 2 Matrix Addition R = A + B A and B must have same size. Result R also has same size In R for all elements, r ij = a ij + b ij Matrix Equality A = B A = B if they are of the same size and corresponding elements are same ie a ij =b ij for all i,j MATLAB Session >> A = [1 3; 2 7]; % define A, dont output it >> B = [8 1; 4 2]; % ditto B >> R = A+B % evaluate sum, assign to R R = >> R = [1 3; 2 7] + [8 1; 4 2]; % dont need to input A and B! Scalar Multiplication R = k*a Each element in A multiplied by a scalar: r ij =k*a ij. Note, k * (A + B) = k*a + k*b 20/09/2013 SEMMA13 Eng Maths and Stats 3 20/09/2013 SEMMA13 Eng Maths and Stats 4 Matrix Multiplication R = A * B Number of columns in A must equal 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, A * B is 2 * 4 matrix. Example Do element 1 of i th row of A *element1ofj th column of B Also do for remaining elements of these rows and columns Find the sum of each of these products and store in r ij MATLAB: >> R = [2 5; 4 3] * [6; 1]; If A * B is ok, then B * A may not be possible. If both are possible, 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) = (A * B) * k {scalar k} 20/09/2013 SEMMA13 Eng Maths and Stats 5 20/09/2013 SEMMA13 Eng Maths and Stats 6 1

2 Example of Use Electronic Example T T N 12V 18Ω 10Ω i 1 i 3 i 2 15Ω By Kirchhoff: 12 = 18 i 1 +10i 2 10 i 2 =15i 3 i 1 =i 2 +i 3 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 This can be expressed in matrices 20/09/2013 SEMMA13 Eng Maths and Stats 7 20/09/2013 SEMMA13 Eng Maths and Stats 8 Matrix Transpose Here the matrix is flipped: rows become columns and vv If R = A T, then r ij = a ji. If A is size m*n, then A T is size n*m. More on Transpose Rules (A T ) T =A (A+B) T =A T +B T (A*B) T =B T *A T (ka) T =ka T MATLAB: >> A 20/09/2013 SEMMA13 Eng Maths and Stats 9 20/09/2013 SEMMA13 Eng Maths and Stats 10 Special Matrices Diagonal Matrices and Equations If S scalar, A * S = S * A. A* I = A To convert a scalar, k, to a matrix, multiply scalar by I Note I n denotes n*n identity matrix, I 3 shown above 20/09/2013 SEMMA13 Eng Maths and Stats 11 It is easy to evaluate - clearly x = 4 y = 5 and z = 9 20/09/2013 SEMMA13 Eng Maths and Stats 12 2

3 Triangular Matrices and Equations Determinants and Inverses Consider the weight suspended by wires problem: It is quite easy to evaluate: Clearly z = 2 from the 3rd row. Then, row 2 gives 2y + 4*z = 18 But z known, so 2y = 18-8 so y = 5. Then, row 1 gives x + 3*5 + 2*2 = 21 so x = 2 One (poor) way is to find inverse or reciprocal of A, A -1. It is defined that A -1 * A = A * A -1 = I the identity matrix and A * I = I * A = A If we know A -1, then by pre-multiplying the equation: A -1 * A * T = A -1 * Y thus T = A -1 * Y How to find the inverse - if one exists (1/0 not exist)? 20/09/2013 SEMMA13 Eng Maths and Stats 13 20/09/2013 SEMMA13 Eng Maths and Stats 14 Determinants, Adjoint and Inverse One way of finding inverse of square matrix Cofactors and Determinants Cofactor C ij of matrix A is -1 (i+j) * M i,j where M i,j is determinant of matrix A without row i and column j Note, a matrix has no inverse if its determinant is 0. The determinant of A is written as det(a) or A The Adjoint of A is denoted Adj(A) Adj(A) is the transpose of the matrix of cofactors of A. If C ij is the cofactor of a ij (defined on next slide), then Adj A, = [C ji ] = [C ij ] T (cofactors use determinants) 20/09/2013 SEMMA13 Eng Maths and Stats 15 The determinant of a single element matrix is that element That of a n*n matrix is the sum of each element in its top row multiplied by that element s cofactor. A = 1 * C * C * C 13 Note: determinants are not just used to find inverses. In fact, determinants are more useful than inverses. 20/09/2013 SEMMA13 Eng Maths and Stats 16 Two by Two Matrices Applied to Suspended Mass Recall Sensible to leave in this form, so divide by 0.8 only twice Check T 1 * T 2 *0.8 = = 0 T 1 * T 2 *0.6 = = /09/2013 SEMMA13 Eng Maths and Stats 17 20/09/2013 SEMMA13 Eng Maths and Stats 18 3

4 Three by Three Matrices Electronic Circuit Example A = 18 * (-10-15) - 10 * (0 --15) + 0 * (0-1) = -600 Ugh! Easiest to remember and calculate by cofactor rule... 20/09/2013 SEMMA13 Eng Maths and Stats 19 20/09/2013 SEMMA13 Eng Maths and Stats 20 Concluded Some Properties (A -1 ) -1 = A (A T ) -1 = (A -1 ) T (A * B) -1 = B -1 * A -1 { note change of order } If A T = A -1 then matrix A is an orthogonal matrix. In MatLab, >> a=[18, 10, 0; 0, -10, 15; -1, 1, 1]; >> v = [12; 0; 0]; > > det(a) ans = -600 >> inv(a)*v ans = For n*n matrix A: A = a 11 * C 11 + a 12 * C a 1n-1 C 1n-1 + a 1n C 1n = a 11 * C 11 -a 12 * M a 1n-1 M 1n-1 + a 1n M 1n (n even) or = a 11 * C 11 -a 12 * M a 1n-1 M 1n-1 -a 1n M 1n (n odd) 20/09/2013 SEMMA13 Eng Maths and Stats 21 20/09/2013 SEMMA13 Eng Maths and Stats 22 Solve Equations Gaussian Elimination Linear equations in examples, in the general form Ax = b. Example Aim to turn 2 in row 2 to 0. Row2 := Row1 * 2 - Row2 * 3 = [ ] Now turn 1 in bottom row to 0 Row3 := Row2 - Row3 * 8 = [ ] Important Each row is equivalent to one equation. Do row operations on each row, row X := rowx*a + rowy*b Aim : make A part of above triangular (so called echelon form) 20/09/2013 SEMMA13 Eng Maths and Stats 23 20/09/2013 SEMMA13 Eng Maths and Stats 24 4

5 So Gauss Jordan for Matrix Inverse To invert A matrix, form matrix [A I], do row operations until A part is unit matrix I, Inverse matrix is in second part Hence, z = -57 / -19 = 3; y = (-1 + 9) / 8 = 1; x = (10-4) / 3 = 2. Zero A(2,1), A(3,1) Zero A(3,2) 20/09/2013 SEMMA13 Eng Maths and Stats 25 20/09/2013 SEMMA13 Eng Maths and Stats 26 Continued MatLab Session Zero A(1,3), A(2,3) Zero A(1,2 Now set diagonals of the left half of the matrix to 1: Divide Row 1 by 360, Row 2 by 120 and Row 3 by -15 >> a = [ 3 4 0; 2 0 1; ]; >> b = [ 10; 7; 7 ]; >> a\b % does GaussElim ans = >> rref ([a b]) % row ops ans = >> aa = [ 3-1 1; ; ]; >> ai = [ aa, eye(3) ] ai = >> rref (ai) ans = /09/2013 SEMMA13 Eng Maths and Stats 27 20/09/2013 SEMMA13 Eng Maths and Stats 28 How many solutions to equations Consider these graphs with 2 lines of form y m x = c y First graph: one value of x and y satisfies both equations, at the intersection of the lines: one solution. Second graph: two lines overlap - infinite solutions. Third graph: two lines are parallel - there is no solution. Any set of linear equations has 0, 1 or solutions x y x y x Finding 0 solutions by G.E. e.g. 2 x + y + 3z = 4; x + y + 2z = 0; 2 x + 4 y + 6 z = 8 Zero the first column from rows 2 and 3 Row2 := Row1-2*Row2 Row3 := Row1-Row3 Row3 := 3*Row2 - Row3 Last row means 0 = 16! So, there is no solution. 20/09/2013 SEMMA13 Eng Maths and Stats 29 20/09/2013 SEMMA13 Eng Maths and Stats 30 5

6 Finding Infinite Solutions by G E e.g. 2 x + y + 3z = 4; x + y + 2z = 0; 2 x + 4 y + 6 z = -8 Eliminating the first column Row2 := Row1-2*Row2 Row3 := Row1-Row3 Rank of Matrix Can determine how many solutions using Rank of a matrix Rank of matrix is largest square submatrix whose det 0. A submatrix of A is A minus some rows or columns Now do Row3 := 3*Row2 - Row3 Last row means 0 = 0; True for all values of x, y and z. solutions to equations. They are linearly dependent. If there are solutions, equations are linearly independent. It has four 3*3 submatrices. All whose det = 0 But, for instance So Rank is 2 20/09/2013 SEMMA13 Eng Maths and Stats 31 20/09/2013 SEMMA13 Eng Maths and Stats 32 Fundamental Theorem of Linear Systems If system defined by m row matrix equation Ax= b The system has solutions only if Rank (A) =Rank(Ã) If Rank (A) = m, one solution If Rank (A) < m, infinite number of solutions Properties of Rank The Rank of A is 0 only if A is the zero matrix. Rank (A) =Rank(A T ) Elementary row operations don't affect matrix rank. Rank is a concept quite useful in control theory. This leads to two related and useful topics Homogeneous Systems If system is Ax= 0, i.e.b =0,systemishomogenous. Suchasystemhasatrivialsolution,x 1 =x 2 =..x n =0. A non trivial solution exists if Rank(A) <m. Cramer's Rule For homogeneous systems If D = A 0, the only solution is x = 0 If D = 0, the system has non trivial solutions This is useful for eigenvalues and eigenvectors. And then there is. 20/09/2013 SEMMA13 Eng Maths and Stats 33 20/09/2013 SEMMA13 Eng Maths and Stats 34 Cramer s Theorem Alternative to Gaussian Elimination Solutions to a linear system A * x = b,(aisan n*n) x 1 =D 1 /D x 2 =D 2 /D... x n =D n /D where D is det A D <> 0 and D k is det of matrix formed by taking A and replacing its kth column with b. Impracticable in large matrices : as hard to find det. Suspended Mass Example Here, D = 0.96 * * 0.28 = 0.8 Replacing first column of A with b, determinant is : Thus T 1 = 240 / 0.8 = 300 N Replacing 2 nd column of A with b, determinant is: Thus T 2 = 288 / 0.8 = 360 N 20/09/2013 SEMMA13 Eng Maths and Stats 35 20/09/2013 SEMMA13 Eng Maths and Stats 36 6

7 Circuit Example D = A found earlier as 600 Other Functions MatLab has commands such as A^3 rank(a) trace(a) exp(a) 20/09/2013 SEMMA13 Eng Maths and Stats 37 20/09/2013 SEMMA13 Eng Maths and Stats 38 Eigenvalues and Eigenvectors Consider the equation Ax= x Here A is n*n matrix, x an n*1 vector and ascalar For a given A we want to find and thence x The n scalars satisfying Ax= x are the eigenvalues of A (or characteristic value or latent root of A) For each, an x satisfying Ax= x is an eigenvector of A Sometimes eigenvalues are labelled as Equation can be reorganised as A * x λ * x = 0 or (A λ * I) x = 0 Homogeneous equation, trivial solution x = 0. By Cramer s Rule, non trivial solution if A - λ*i = 0 n eigenvalues found by solving characteristic equation A - λ*i = 0 2*2 Example A - λ*i =(-4-λ)(-3 - λ) -2*1 = λ 2 +4λ +3λ = λ 2 +7λ +10 In this case, the characteristic equation is λ 2 +7λ +10=0 Characteristic polynomial is λ 2 +7λ +10 = (λ +5)(λ +2) The solutions for the characteristic equation are λ =-5andλ =-2 These are the two eigenvalues for the matrix A. 20/09/2013 SEMMA13 Eng Maths and Stats 39 20/09/2013 SEMMA13 Eng Maths and Stats 40 Finding Eigenvectors of A Other Eigenvector Eigenvectors of A are x satisfying (A - λ*i) x = 0 for each λ This represents equations: x 1 +x 2 =0 and 2*x 1 +2*x 2 =0 Not independent - infinitely many solutions - always happens This represents : -2*x 1 +x 2 =0 and 2*x 1 -x 2 =0 Again not independent: let x 1 =1, so x 2 =2. So eigenvector So choose x 1 =1, say, then x 2 =-1. Thus an eigenvector is As are all multiples of this vector (recall Ax = λx) 20/09/2013 SEMMA13 Eng Maths and Stats 41 20/09/2013 SEMMA13 Eng Maths and Stats 42 7

8 Application - Diagonalisation To diagonalise an n*n matrix A. Let its eigenvalues and eigenvectors are λ 1, λ 2,.. λ n, and Λ 1, Λ 2,.. Λ n. Let X = [Λ 1 Λ 2.. Λ n ], then A- λi = (5 - λ)(2 - λ) - 4 = 0 or λ 2-7λ + 6 = 0 Eigenvalues are 1 and 6. MatLab Session >> a = [-4 1; 2-3]; >> [evec eval] = eig(a) evec = % Note, MATLAB arranges that % eigenvector is normalised % (sum of squares of eval = % eigenvector values set to 1) -5 0 % eigenvalues down diagonal 0-2 >> inv(evec)*a*evec % to show diagonalisation /09/2013 SEMMA13 Eng Maths and Stats 43 20/09/2013 SEMMA13 Eng Maths and Stats 44 Singular Values For an n*n matrix A. Form the symmetric matrix A T A Find the eigenvalues of this matrix, λ 1, λ 2,.. λ n The singular values are λ r These are used in Principal Component Analysis, Singular Value Decomposition, etc Positive Definite Matrices Suppose A is a symmetric n x n matrix, A = A T. For any non zero column vector x, containing n real numbers A is Positive Definite if x T Ax is positive (> 0) It is Positive Semi Definite if x T Ax ³ 0 A is positive definite if all its eigenvalues are positive. A symmetric matrix is positive definite if: all the diagonal entries are positive, and each diagonal entry is greater than the sum of the absolute values of all other entries in the corresponding row/column. If x T Ax is negative, A is negative definite (its eigenvalues all < 0) 20/09/2013 SEMMA13 Eng Maths and Stats 45 20/09/2013 SEMMA13 Eng Maths and Stats 46 Vectors A scalar has magnitude: e.g. mass, speed, temperature, position A vector has magnitude and direction: e.g. force, velocity Can define position in (x,y,z) terms, e.g. P = (1, 2, 5), Q = (4, -2, 1) A vector defines movement from one position to another Vector P to Q has components 4-1 in x, -2-2 in y and 1-5 in z In general, if P is point (x 1,y 1,z 1 )andqis(x 2,y 2,z 2 ) then the vector from P to Q, is (x 2 -x 1,y 2 -y 1,z 2 -z 1 ) Some Notation A vector is a row/column matrix: eg for 3D v =[xyz] For 2 dimensions, a vector has 2 components v =[xy] Can extend concept to n dimensions v =[d 1 d 2.. d n ] The magnitude (or length) of vector v is given by n dimensional Pythagoras e.g. v = [3 4] v = (9 + 16) = 5 v = [3 4 12] v = ( ) = 13 20/09/2013 SEMMA13 Eng Maths and Stats 47 20/09/2013 SEMMA13 Eng Maths and Stats 48 8

9 Norm s of vectors This is the vector length, sometimes called the Euclidean distance This is another norm, sometimes called Taxicab or Manhattan norm This is the L p norm (p 1 and real) If p = 1, is Manhattan norm If p = 2, is Euclidean norm Matrix Norm For n*m matrix A and m*1 vector x, the p-norm of A is Here sup is the supremum or least upper bound For p is 1, the norm is the maximum of the sum of each column in A For p is 2, the norm is the largest singular value of A 20/09/2013 SEMMA13 Eng Maths and Stats 49 20/09/2013 SEMMA13 Eng Maths and Stats 50 Vectors in 2D P to Q = (4-1, -2-2) = (3, 4) Q to P = (1-4, 2 2) = (-3, 4) = -(P to Q) PtoQthenQtoRequivtoPtoR Suggests a + b = c P=(p x, p y )Q=(q x,q y ) R = (r x,r y ) a = [q x -p x q y -p y ], b = [r x -q x r y -q y ] c =[r x -p x r y -p y ] a + b = [q x -p x + r x -q x q y -p y + r y -q y ] = [r x -p x r y -p y ] = c Orthogonal vectors A convenient representation is to use i,j (and k) These have length 1 and are at right angles to each other i = [1 0] or [1 0 0] j = [0 1] or [0 1 0] k = [0 0 1] Clearly unit length: i = j = k = 1 In use: v = [x y] = xi + yj e.g. v =[23]=2i +3j In 3D: v =[xyz]=xi +yj +zk 20/09/2013 SEMMA13 Eng Maths and Stats 51 20/09/2013 SEMMA13 Eng Maths and Stats 52 Dot products Vector Cross Product The dot or inner product of two vectors a and b is a scalar: a.b = a b cos(θ) where θ is angle between a and b For 2 orthogonal vectors ( = 90 O ), their dot product is 0: i. j = 0 a. b = (a 1 i + a 2 j). (b 1 i + b 2 j) = a 1 i. b 1 i + a 1 i. b 2 j + a 2 j. b 1 i + a 2 j. b 2 j = a 1 *b 1 + a 2 *b 2 e.g. [2 3]. [5 7] = 2*5 + 3*7 = 31 In general, if a = [ a 1 a 2... a n ] and b = [ b 1 b 2... b n ] then a.b = a 1 *b 1 + a 2 *b a n *b n a θ b Vector cross product of 2 vectors is also a vector: v=ax b v is a vector that is at right angles to both a and b Length of v is defined as a b sin(θ) where θ is angle between a and b The components of v are [a 2 b 3 -a 3 b 2 a 3 b 1 -a 1 b 3 a 1 b 2 -a 2 b 1 ] Tricky to remember but there is a trick from Matrices v θ a b 20/09/2013 SEMMA13 Eng Maths and Stats 53 20/09/2013 SEMMA13 Eng Maths and Stats 54 9

10 Vector Products and Determinants Examples Suppose Force F applied to object moving it a distance d is angle between direction of movement and of F Work done = (length of d) * (component of F in direction d) This is: d * F cos ( ) = F.d i x j = k j x k = i k x i = j j x i =-k k x j =-i k x i =-j Scalar Triple Product e.g. F = [3 4], d = [6 0]. Work done = 3*6 + 4*0 = 18 Geometric applications exist for these products 20/09/2013 SEMMA13 Eng Maths and Stats 55 20/09/2013 SEMMA13 Eng Maths and Stats 56 r Another Example f Force f applied to object at Q. What is moment of force on object about P? If force f is applied at distance d, & f to d, moment is f * d Required force is f sin( ); distance at which it acts is r Therefore the moment is r f sin( ) =r f e.g. P is at (3,2,4), Q is at (7,3,6) and f =[431] So r =4i +1j +2k and f =4i +3j +1k Back Face Elimination 2 vectors on surface vector normal to surface surface 'normal' vector Eye Angle ~120 Thus surface visible In Computer Graphics, object has surfaces Display surfaces of objects which face eye find by Is vector normal (perpendicular) to surface pointing towards eye? vs 1, vs 2 are vectors on surface vn = vs 1 x vs 2 is normal to surface ve is vector from eye to the surface Surface visible if angle between ve and vn < -90 O or > 90 O Then cos(angle) < 0, so ve.vn < 0 NB ve.vn = ve.(vs 1 x vs 2 ) i.e. a triple scalar product 20/09/2013 SEMMA13 Eng Maths and Stats 57 20/09/2013 SEMMA13 Eng Maths and Stats 58 Vector Space and Sets A linear vector space is a set V whose elements are vectors on whom addition and multiplication can act. So if vectors x and y are in V : x, y Î V, and a and b are real numbers: x + y Î V and x + y = y + x a zero vector 0 Î V exists such that for all x Î V, x + 0 Î V for x Î V, a * x Î V for x,y Î V, a (x + y) = ax + ay; (a + b)x = ax + bx These properties apply to  n, n dimensions of real numbers R A subset U of  n, denoted U Ì Â n is open if for all vectors x Î U a neighbourhood, y, of x can be found, within a distance d. More on Sets A point x is a boundary point of a subset U Ì Â n if every neighbourhood of x contains at least one point belonging to U and one not. The boundary of U is the set of all boundary points of U, denoted U Point x is a boundary point of a subset U Ì n if every neighbourhood of x contains at least one point belonging to U and one not. The boundary of U is the set of all boundary points of U, denoted U The interior of a set A is A - A The closure of set, Ā is the union of A and its boundary An open set is equal to its interior, a closed set is equal to its closure A subset U Ì n is bounded if there is a r > 0 such that A set is closed if its complement in  n is an open set 20/09/2013 SEMMA13 Eng Maths and Stats 59 A set S Ì Â n is compact if it is closed and bounded 20/09/2013 SEMMA13 Eng Maths and Stats 60 10

11 SEMMA13 Complex Numbers Consider solutions of quadratic equation x 2 + 4x + 13 = 0 In general complex number z, has the form a + j b a is the real part a = Re(z) b is the imaginary part b = In the above example have -2 + j3 and -2 j3 The latter is the complex conjugate of the former If z = a + jb its complex conjugate denoted z is a - jb. Rules on Complex Numbers z = a + jb and w = c + jd are equal only if a = c and b = d. z + w = (a + c) + j(b + d) z - w = (a - c) + j(b - d) z * w = (a + jb)(c + jd)= ac + ajd + jbc + j 2 bd But j 2 = -1 so that z * w = ac + ajd + jbc - bd= ac - bd + j(ad + bc) z * z = (a + jb)(a - jb) = a 2 -a(jb) + (jb)a - j 2 b 2 = a 2 - j 2 b 2 = a 2 + b 2 This is always a real number 0 20/09/2013 SEMMA13 Eng Maths and Stats 61 20/09/2013 SEMMA13 Eng Maths and Stats 62 On Division Graphical Representation Normal numbers represented by point on infinite line - to + So we multiply top and bottom by the complex conjugate of the denominator, by c - jd (this is called rationalising): Complex numbers represent on infinite 2D plane Argand plane horiz axis for real part, vertical axis for imaginary 5 This is Cartesian representation In many practical applications of complex numbers, we don t need to do this, as we shall see 20/09/2013 SEMMA13 Eng Maths and Stats 63 This leads to an alternative way of representing complex numbers 20/09/2013 SEMMA13 Eng Maths and Stats 64 Polar Representation Use r, instead of a,b for same complex number b r a z = a + j b Re(z) r, modulus, z is distance from origin, argument, z, is angle from real axis Complex Number can be represented by a and b or r and Can interchange between Cartesian and Polar forms b b+d Adding on Argand Plane z = a + j b a Re(z) z = a + j b w = c + j d a+c d Re(z) w = c + j d c Re(z) a + j b + c + j d = a+c + j(b+d) Adding easy in Cartesian form, more difficult in Polar form 20/09/2013 SEMMA13 Eng Maths and Stats 65 20/09/2013 SEMMA13 Eng Maths and Stats 66 11

12 Multiplying and Dividing Multiplying in Polar Form b z = r a Re(z) d w = s c Re(z) r*s + If want modulus and argument, easier to multiply in polar form + Division also easier : don t need to use conjugate etc z = r Re(z) Multiplying easy in Polar form 20/09/2013 SEMMA13 Eng Maths and Stats 67 20/09/2013 SEMMA13 Eng Maths and Stats 68 Exponential Form It can be shown, see later on Taylors series, that Hyperbolic and Trigonometric Fn e jθ = cos θ + j sin θ and e -jθ = cos(-θ) + j sin (-θ), add/sub gives: But So Exponential Form useful in various integrals 20/09/2013 SEMMA13 Eng Maths and Stats 69 20/09/2013 SEMMA13 Eng Maths and Stats 70 De-Moivre s Theorem (e jθ ) p = e jpθ suggesting that (cos θ + jsin θ) p = (cos pθ + jsin pθ) Fine where p is an integer, but when p is a fraction be careful As cos θ = cos (θ+2π) = cos (θ+2kπ) for all integer k So e jθ = e j(θ+2kπ) Complex Matrices Matrices can have complex numbers as elements As such can have conjugate matrix The eigenvalues of a Hermitian matrix are real; Those of a skew-hermitian matrix are imaginary or zero If eigenvalues of a matrix are complex (conjugate pairs), so are its eigenvectors 20/09/2013 SEMMA13 Eng Maths and Stats 71 20/09/2013 SEMMA13 Eng Maths and Stats 72 12