Math 211 Math 211 Complex Numbers and Matrices October 29, 2001 2 - PowerPoint PPT Presentation

1 Math 211 Math 211 Complex Numbers and Matrices October 29, 2001

2 Complex Numbers Complex Numbers A complex number is one of the form z = x + iy , where x and y are real numbers. • Geometric representation — the complex plane. � z = x + iy ↔ ( x, y ) . • x is the real part of z ; x = Re z . • y is the imaginary part of z ; y = Im z . � The imaginary part of the complex number z = x + iy is the real number y . • Addition and multiplication ( i 2 = − 1 ). Return

3 Complex Conjugate Complex Conjugate The conjugate of z = x + iy is z = x − iy. Definition: • z = z ⇔ z is a real number. • x = Re z = z + z ; y = Im z = z − z 2 2 i • z + w = z + w ; z − w = z − w � z = z � • zw = z · w ; w w Return

4 Absolute Value Absolute Value The absolute value of z = x + iy is the real Definition: x 2 + y 2 . � number | z | = • z · z = | z | 2 = x 2 + y 2 . • | zw | = | z || w | � z � = | z | � � • � � w | w | • | z + w | ≤ | z | + | w | Return

5 Quotients Quotients • The reciprocal of z = x + iy 1 z = 1 z · z z = z z zz = | z | 2 . x + iy = x − iy 1 x 2 + y 2 • The quotient w = z · 1 z | w | 2 = zw w w = z · | w | 2 Return zz

6 Polar Representation Polar Representation • z = x + iy = r [cos θ + i sin θ ] . � θ is called the argument of z . ◮ tan θ = y/x. � r = | z | . • Euler’s formula: e iθ = cos θ + i sin θ. � z = | z | e iθ . � z = | z | e − iθ . Return

7 Multiplication Multiplication • Two complex numbers z = | z | e iθ w = | w | e iφ and • The product is zw = | z | e iθ · | w | e iφ = | z || w | e i ( θ + φ ) . � The absolute value of the product zw is the product of the absolute values of z and w : | zw | = | z || w | . � The argument of the product zw is the sum of the arguments of z and w . Return

8 Complex Exponential Complex Exponential For z = x + iy we define Definition: e z = e x + iy = e x · e iy = e x [cos y + i sin y ] . Properties: • e z + w = e z · e w ; e z − w = e z · e − w = e z /e w • e z = e z • | e z | = e x = e Re z • If λ is a complex number, then d dte λt = λe λt Return

9 Complex Matrices Complex Matrices Matrices (or vectors) with complex entries inherit many of the properties of complex numbers. • M = A + iB where A = Re M and B = Im M are real matrices. • M = A − iB ; M = M ⇔ M is real. • Re M = 1 Im M = 1 2 ( M + M ) ; 2 i ( M − M ) • M + N = M + N • M z = M z Return