1. ENGG2013 Unit 20 Extensions to Complex numbers Mar, 2011.

2. Definition: Norm of a vector By Pythagoras theorem, the length of a vector with two components [a b] is The length of a vector with three components [a b c] is The length of a vector with n components, [a 1 a 2 … a n ], is defined as, which is also called the norm of [a 1 a 2 … a n ]. kshumENGG20132

3. Examples We usually denote the norm of a vector v by || v ||. kshumENGG20133

4. Norm squared The square of the norm, or square of the length, of a column vector v can be conveniently written as the dot product Example kshumENGG20134

5. REVIEW OF COMPLEX NUMBERS kshumENGG20135

6. Quadratic equation When the discriminant of a quadratic equation is negative, there is no real solution. The complex roots are kshumENGG20136

7. Complex eigenvalues There are some matrices whose eigenvalues are complex numbers. The characteristic polynomial of this matrix is The eigenvalues are kshumENGG20137

8. Complex numbers Let i be the square root of –1. A complex number is written in the form a+bi where a and b are real numbers. a is called the real part and b is called the imaginary part of a+bi. Addition: (1+2i) + (2 – i) = 3+i. Subtraction: (1+2i) – (2 – i) = –1 + 3i. Multiplication: (1+2i)(2 – i) = 2+4i–i–2i 2 =4+3i. kshumENGG20138

9. Complex numbers The conjugate of a+bi is defined as a – bi. The absolute value of a+bi is defined as (a+bi)(a – bi) = (a 2 +b 2 ) 1/2. – We use the notation | a+bi | to stand for the absolute value a 2 +b 2. Division: (1+2i)/(2 – i) kshumENGG20139

10. The complex plane kshumENGG201310 Re Im 1+2i 2 – i 3+i

11. Polar form kshumENGG201311 Re Im a+bi = r (cos + i sin ) = r e i r a b

12. COMPLEX MATRICES kshumENGG201312

13. Complex vectors and matrices Complex vector: vector with complex entries – Examples: Complex matrix: matrix with complex entries kshumENGG201313

14. Length of complex vector If we apply the calculation of the length of a vector to a complex, something strange may happen. – Example: the length of [i 1] would be – Example: the length of [2i 1] would be kshumENGG201314

15. Definition The norm, or length, of a complex vector [z 1 z 2 … z n ] where z 1, z 2, … z n are complex numbers, is defined as Example – The norm of [i 1] is – The norm of [2i 1] is kshumENGG201315

16. Complex dot product For complex vector, the dot product is replaced by where c 1, d 1, e 1, c 2, d 2, e 2 are complex numbers and c 1 *, d 1 *, and e 1 * are the conjugates of c 1, d 1, and e 1 respectively. kshumENGG201316

17. The Hermitian operator The transpose operator for real matrix should be replaced by the Hermitian operator. The conjugate of a vector v is obtained by taking the conjugate of each component in v. The conjugate of a matrix M is obtained by taking the conjugate of each entry in M. The Hermitian of a complex matrix M, is defined as the conjugate transpose of M. The Hermitian of M is denoted by M H or. kshumENGG201317

18. Example kshumENGG201318 Hermitian

19. Example kshumENGG201319

20. Complex matrix in special form Hermitian: A H =A. Skew-Hermitian: A H = –A. Unitary: A H =A -1, or equivalently A H A = I. Example: kshumENGG201320

21. Charles Hermite Dec 24, 1822 – Jan 14, 1901. French mathematician Introduced the notion of Hermitian operator Proved that the base of the natural log, e, is transcendental. kshumENGG201321 http://en.wikipedia.org/wiki/Charles_Hermite

22. Properties of Hermitian matrix Let M be an n n complex Hermitian matrix. The eigenvalues of M are real numbers. We can choose n orthonormal eigenvectors of M. – n vectors v 1, v 2, …, v n, are called orthonormal if they are (i) mutually orthogonal v i H v j =0 for i j, and (ii) v i H v i =1 for all i. We can find a unitary matrix U, such that M can be written as UDU H, for some diagonal matrix with real diagonal entries. kshumENGG201322 http://en.wikipedia.org/wiki/Hermitian_matrix

23. Properties of skew-Hermitian matrix Let S be an n n complex skew-Hermitian matrix. The eigenvalues of S are purely imaginary. We can choose n orthonormal eigenvectors of S. We can find a unitary matrix U, such that S can be written as UDU H, for some diagonal matrix with purely imaginary diagonal entries. kshumENGG201323 http://en.wikipedia.org/wiki/Skew-Hermitian_matrix

24. Properties of unitary matrix Let U be an n n complex unitary matrix. The eigenvalues of U have absolute value 1. We can choose n orthonormal eigenvectors of U. We can find a unitary matrix V, such that U can be written as VDV H, for some diagonal matrix whose diagonal entries lie on the unit circle in the complex plane. kshumENGG201324 http://en.wikipedia.org/wiki/Unitary_matrix

25. Eigenvalues of Hermitian, skew- Hermitian and unitary matrices kshumENGG201325 Hermitian Re Im Complex plane 1 Skew-Hermitian unitary

26. Generalization: Normal matrix A complex matrix N is called normal, if N H N = N N H. Normal matrices contain symmetric, skew- symmetric, orthogonal, Hermitian, skew- Hermitain and unitary as special cases. We can find a unitary matrix U, such that N can be written as UDU H, for some diagonal matrix whose diagonal entries are the eigenvalues of N. kshumENGG201326 http://en.wikipedia.org/wiki/Normal_matrix

27. COMPLEX EXPONENTIAL FUNCTION kshumENGG201327

28. Exponential function Definition for real x: kshumENGG201328 y = e x. http://en.wikipedia.org/wiki/Exponential_function

29. Derivative of exp(x) kshumENGG201329 y= e x y=1+x For example, the slope of the tangent line at x=0 is equal to e 0 =1.

30. Taylor series expansion We extend the definition of exponential function to complex number via this Taylor series expansion. For complex number z, e z is defined by simply replacing the real number x by complex number z: kshumENGG201330

31. Series expansion of sin and cos Likewise, we extend the definition of sin and cos to complex number, by simply replacing real number x by complex number z. kshumENGG201331 http://en.wikipedia.org/wiki/Taylor_series

32. Example For real number : kshumENGG201332

33. Eulers formula kshumENGG201333 For real number, Proof:

34. Summary Matrix and vector are extended from real to complex – Transpose conjugate transpose (Hermitian operator) – Symmetric Hermitian – Skew-symmetric skew-Hermitian Exponential function and sinusoidal function are extended from real to complex by power series. kshumENGG201334